# Andriy Zhugayevych, azh@ukr.net
# Finite Groups package
# created 2.10.2004, modified - see version below
#
################################################################################
#root: FiniteGroups
#hfl: FiniteGroups
#toc: _Overview
FiniteGroups:=
module() option package;

global FG_simplify, FG_floor;

export ModuleLoad, Setup, setfloordef, setfloorfast, setfloorback,
       FG_GroupAlgebra, FGS, FG_unit, FG_ident, FG_inv, FG_mul, FG_app, FG_appt, FG_apl,
        unitcellon, unitcelloff, unitcellback, # Group algebra
       FG_member, FG_in, FG_equal, FG_set, FG_prod, FG_transform, # Manipulations with sets
       FG_test, FG_testnumeric, FG_group, FG_mulTable, FG_invTable, FG_mulinvTable,
        FG_cyclic, FG_orbitC, FG_factor, FG_testInvSG, FG_testSDP, FG_testDP, # Manipulations with groups
       FG_orbit, FG_RegRep, # Representations of groups
       FGR_ChTb, FGR_RepU, FGR_RepU2, FGR_RepDec, FGR_Ch, FGR_RepDecCh, FGR_indexC, FGR_dimR, # Representations of predefined groups
       FGS_init, FGS_get, FGS_correctit, FGS_correctost, FGS_it2g, FGS_g2it, FGS_inv, FGS_mul,
        FGS_s2it, FGS_app, FGS_s2m, FGS_m2s, FGS_p2s, FGS_s2p, FGS_m2p, FGS_spick,
        FGS_d2oost, FGS_d2oost1, FGS_d2oost2, FGS_dorb, FGS_dorb2, FGS_dpick,
        FGS_dimersR, FGS_dimersR2, Encodesoost, Decodesoost, # Sites and dimers
        FGS_ReduceDimer, FGS_ReduceDimer2, FGS_UnfoldDimer2, # Sites and dimers (obsolete code)
       PrintSymmetryElement, SymmetryGroup, kPoints, HM2Scho, Scho2HM,
        PointGroupName, SpaceGroupName, LayerGroupName, RodGroupName, OtherGroupName,
        SpaceGroupNumber, LayerGroupNumber, RodGroupNumber,
        ArithmeticClass, GeometricClass, TransformationTable; # Tables of groups

local  version, parvalues, setting, onpd, origin, unitcell, unitcell9, floor9, reduce0, reduce, transl, i, j, k, v, f, tbG, lsG, tbK,
       sites_SS, sites_floor, sites_pcoo, sites_wcoo, sites_table, sites_SG, sites_G, sites_ng, sites_affine,
        sites_nd, sites_setting, sites_origin, sites_npd, sites_id, sites_P, sites_P0, sites_orb, sites_t0,
        sites_gen, sites_stab1, sites_stab, sites_oi2s, sites_ois2s, sites_ois2t,
        sites_m2os, sites_os2m, sites_no, sites_ns, sites_nm,
       dimers_orb, dimers_id, dimers_sort;


ModuleLoad:=proc()
local l,v,u,w;
version:=20231123;
# BasicTools package is required
if not(member('BasicTools',packages())) then try with(BasicTools) catch: error("Cannot find the required BasicTools package: %1",lastexception) end end;
if (BasicTools[Setup]('version')<20221030) then WARNING("Update BasicTools package to version from 30.10.2022 at least") end;
# Default settings
FG_simplify:=simplify@evalc;
parvalues:=[seq(exp(-2.07-0.5*l),l=1..32)];
setfloordef();
Setup('setting'="affine",'origin'=[0,0,0]);
for v in indices(TransformationTable,'nolist') do
 u:=TransformationTable[v];
 if (type(u,string) and not(assigned('tbG[u]'))) then
  w:=[SymmetryGroup(u,'output'="nT",'nosetup')];
  if (nops(w)=2 and type(w[1],string)) then TransformationTable[v]:=w end end end;
for v in indices(tbG,'nolist') do if type(tbG[v],string) then
 w:=SymmetryGroup(tbG[v],'nosetup');
 if type(w,list) then tbG[v]:=w else error("tbG initialization error: v=%1, u=%2, w=%3",v,u,w) end end end;
for v in indices(TransformationTable,'nolist') do
 u:=TransformationTable[v];
 if (type(u,string) and not(assigned('tbG[u]'))) then
  w:=[SymmetryGroup(u,'output'="nT",'nosetup')];
  if (nops(w)=2 and type(w[1],string)) then TransformationTable[v]:=w else error("TransformationTable initialization error: v=%1, u=%2, w=%3",v,u,w) end end end;
Setup('setting'="matrix",'origin'=[9,9,9],'onpd'=9,'unitcell'=false);
NULL
end:

reduce0:=x->x-floor(x):

reduce:=proc(p0,origin) local p,k;  p:=copy(p0);  for k from 1 to nops(origin) do if (origin[k]<>onpd) then p[k]:=p[k]-FG_floor(p[k]-origin[k]) end end;  p end:

transl:=proc(p0,origin) local p,k;  p:=copy(p0);  for k from 1 to nops(origin) do p[k]:=`if`(origin[k]=onpd,0,-FG_floor(p[k]-origin[k])) end;  p end:


#hfl: FiniteGroups
setfloordef :=proc()  floor9:=op(FG_floor);  FG_floor:=proc(x) local e,l;  e:=indets(x);  floor(eval(x,{seq(e[l]=parvalues[l],l=1..nops(e))})) end;  NULL end:

#hfl: FiniteGroups
setfloorfast:=proc()  floor9:=op(FG_floor);  FG_floor:=floor;  NULL end:

#hfl: FiniteGroups
setfloorback:=proc()  FG_floor:=op(floor9);  NULL end:



#hfl: Setup
#toc: _Setup
#FiniteGroups[Setup]
Setup:=proc({printout::boolean:=false})
local setting9,sq;
setting9:=setting;
sq:=ProcessSetupArgs([_rest],['setting','origin','onpd','unitcell'],['version']);
if member("setting =",map(v->convert(v,string)[..9],[_rest])) then
 if (setting="Matrix") then setting:="matrix"; WARNING("Use 'matrix' instead of 'Matrix'") elif (setting="Space") then setting:="affine"; WARNING("Use 'affine' instead of 'Space'") end; #for backward compatibility
 if not(member(setting,["matrix","affine"])) then setting:=setting9; error("Unrecognized setting=%1",setting) end;
 FG_GroupAlgebra(setting) end;
if printout then printf("setting=%s, origin=[%{c,}a], unitcell=%a\n",setting,Vector(origin),unitcell) end;
sq
end:





################################################################################
#cat: Group algebra
#hfl: GroupAlgebra
#toc: _Overview
#Group algebra

#hfl: GroupAlgebra
FG_GroupAlgebra:=proc(s::{"matrix","affine"},$)
setting:=s;
unprotect('FGS,FG_unit,FG_ident,FG_inv,FG_mul,FG_app,FG_appt,FG_apl');
if (setting="matrix") then
 FGS:=g->map(FG_simplify,g);
 FG_unit:=g->LinearAlgebra[IdentityMatrix](op(1,g));
 FG_ident:=g->andmap(`=`,FGS(`if`(_npassed=1,g,g-_passed[2])),0);
 FG_inv:=g->FGS(g^(-1));
 FG_mul:=(g1,g2)->FGS(g1.g2);
 FG_app:=(g,p)->FGS(g.p);
 FG_appt:=(g,p)->Vector(op(1,p));
 FG_apl:=(g,lsp)->map(p->FGS(g.p),lsp)
elif (setting="affine") then
 FGS:=g->`if`(type(g,[Matrix,Vector]),[map(FG_simplify,g[1]),map(`if`(unitcell,reduce0@FG_simplify,FG_simplify),g[2])],map(FG_simplify,g));
 FG_unit:=(g::{Matrix,[Matrix,Vector]})->`if`(type(g,Matrix),LinearAlgebra[IdentityMatrix](op(1,g)),[LinearAlgebra[IdentityMatrix](op(1,g[1])),Vector(op(1,g[2]))]);
 FG_ident:=proc(g) local m1,v1,m2,v2;
  if type(g,[Matrix,Vector]) then m1,v1:=op(g) elif type(g,Matrix) then m1,v1:=g,0 elif type(g,Vector) then m1,v1:=0,g else error("Unrecognized g: %1",g) end;
  if (_npassed=1) then m2,v2:=0,0 elif type(_passed[2],[Matrix,Vector]) then m2,v2:=op(_passed[2]) elif type(_passed[2],Matrix) then m2,v2:=_passed[2],0 elif type(_passed[2],Vector) then m2,v2:=0,_passed[2] else error("Unrecognized g2: %1",_passed[2]) end;
  andmap(`=`,FGS(`if`(m1=0,m2,`if`(m2=0,m1,m1-m2))),0) and andmap(`=`,FGS(`if`(v1=0,v2,`if`(v2=0,v1,v1-v2))),0) end:
 FG_inv:=(g::{Matrix,[Matrix,Vector]})->FGS(`if`(type(g,Matrix),g^(-1),[g[1]^(-1),-g[1]^(-1).g[2]]));
 FG_mul:=(g1::{Matrix,[Matrix,Vector]},g2::{Matrix,[Matrix,Vector]})->FGS(`if`(type(g1,Matrix),
  `if`(type(g2,Matrix),g1.g2,[g1.g2[1],g1.g2[2]]),
  `if`(type(g2,Matrix),[g1[1].g2,g1[2]],[g1[1].g2[1],g1[1].g2[2]+g1[2]])));
 FG_app:=proc(g::{Matrix,[Matrix,Vector]},p::Vector,origin2::list:=origin,unitcell2::boolean:=unitcell,$) local p2; # $ must be removed if parameters are inconsistent with other settings
   p2:=FGS(`if`(type(g,Matrix),g.p,g[1].p+g[2]));  `if`(unitcell2,reduce(p2,origin2),p2) end;
 FG_appt:=proc(g::{Matrix,[Matrix,Vector]},p::Vector,origin2::list:=origin,$) local p2;
   p2:=FGS(`if`(type(g,Matrix),g.p,g[1].p+g[2]));  transl(p2,origin2) end;
 FG_apl:=proc(g::{Matrix,[Matrix,Vector]},lsp::list(Vector),origin2::list:=origin,unitcell2::boolean:=unitcell,$) local p2,t;
  if type(g,Matrix) then map(p->FGS(g.p),lsp) else  t:=`if`(unitcell2,g[2]+transl(FGS(g[1].(add(p2,p2=lsp)/nops(lsp))+g[2]),origin2),g[2]);  map(p->FGS(g[1].p+t),lsp) end end
end;
protect('FGS,FG_unit,FG_ident,FG_inv,FG_mul,FG_app,FG_appt,FG_apl')
end:

#hfl: GroupAlgebra
#FGS(g)
#hfl: GroupAlgebra
#FG_unit(g)
#hfl: GroupAlgebra
#FG_ident
#hfl: GroupAlgebra
#FG_inv(g)
#hfl: GroupAlgebra
#FG_mul
#hfl: GroupAlgebra
#FG_app
#hfl: GroupAlgebra
#FG_apl

#hfl: GroupAlgebra
unitcellon  :=proc() unitcell9:=unitcell; unitcell:=true end:

#hfl: GroupAlgebra
unitcelloff :=proc() unitcell9:=unitcell; unitcell:=false end:

#hfl: GroupAlgebra
unitcellback:=proc() unitcell:=unitcell9 end:





################################################################################
#cat: Manipulatiions with sets
#hfl: Sets
#toc: _Overview
#Manipulatiions with sets

#hfl: Sets
FG_member:=proc(g,S::list,$)
local i;
for i from nops(S) by -1 to 1 do if FG_ident(g,S[i]) then break end end;
i
end:

#hfl: Sets
FG_in:=proc(SS::list,S::list,$)
local g,b;
b:=true;
for g in SS do if (FG_member(g,S)=0) then b:=false; break end end;
b
end:

#hfl: Sets
FG_equal:=(S1::list,S2::list)->evalb(nops(S1)=nops(S2) and FG_in(S2,S1) and FG_in(S1,S2)):

#hfl: Sets
FG_set:=proc(preS::list,$)
local S,i;
S:=preS;
i:=2;
while (i<=nops(S)) do if (FG_member(S[i],S[1..i-1])>0) then S:=subsop(i=NULL,S) else i:=i+1 end end;
S
end:

#hfl: Sets
FG_prod:=proc(S1::list,S2::list,$) local g1,g2; [seq(seq(FG_mul(g1,g2),g1=S1),g2=S2)] end:

#hfl: Sets
FG_transform:=proc(S::list,T::{Matrix,[Matrix,Vector],Vector},$)
local d,TM,TV,TMi,S2;
if (S=[]) then return [] end;
d:=op(1,`if`(type(S[1],list),S[1][1],S[1]))[1];
TM,TV:=LinearAlgebra[IdentityMatrix](d),Vector(d);
if type(T,list) then TM,TV:=op(T) elif type(T,Matrix) then TM:=T else TV:=T end;
TMi:=FGS(TM^(-1));
if type(S[1],list) then
 unitcellon();
 S2:=map(g->FGS([TMi.g[1].TM,TMi.(g[1].TV-TV+g[2])]),S);
 unitcellback()
else
 S2:=map(g->FGS(TMi.g.TM),S) end;
S2
end:





################################################################################
#cat: Manipulations with groups
#hfl: Groups
#toc: _Overview
#Manipulations with groups

#hfl: Groups
FG_test:=proc(S::list,{timing::boolean:=false},$)
local g1,g2;
if (S=[] or FG_member(FG_unit(S[1]),S)=0) then return false end;
for g1 in S do if (FG_member(FG_inv(g1),S)=0) then return false end end;
if timing then InitTiming(nops(S)) end;
for g1 in S do for g2 in S do if (FG_member(FG_mul(g1,g2),S)=0) then return false end end;  if timing then Timing() end end;
true
end:

#hfl: Groups
FG_testnumeric:=proc(S::list,p0::Vector,$)
local lsp,i,j,g,p,p2,flag,p3;
if (S=[] or FG_member(FG_unit(S[1]),S)=0) then return false end;
lsp:=map(FG_app,G,p0);
for i from 1 to nops(lsp) do for j from i+1 to nops(lsp) do if verify(lsp[i],lsp[j],Vector) then error("Non-generic initial point: lsp[%1]=lsp[%2], p0=%3",i,j,p0) end end end;
InitTiming(nops(S));
for g in G do for p in lsp do
 p2:=FG_app(g,p);
 flag:=false;
 for p3 in lsp do if verify(p2,p3,Vector) then flag:=true; break end end;
 if not(flag) then return false end end;  Timing() end;
true
end:

#hfl: Groups
FG_group:=proc(S::list,nmax::posint:=2*nops(S),$)
local E,G,n,g,g1,g2;
unitcellon();
E:=FG_unit(S[1]);
if (FG_member(E,S)=0) then G:=[E,op(S)] else G:=S end;
n:=1;
while (nops(G)>n) do
 n:=nops(G);
 for g1 in G do
  g:=FG_inv(g1);
  if (FG_member(g,G)=0) then G:=[op(G),g] end;
  if (nops(G)>nmax) then error("The group seems to be infinite, nmax=%1",nmax) end end;
 for g1 in G do for g2 in G do
  g:=FG_mul(g1,g2);
  if (FG_member(g,G)=0) then G:=[op(G),g] end;
  if (nops(G)>nmax) then error("The group seems to be infinite, nmax=%1",nmax) end end end end;
unitcellback();
G
end:

#hfl: Groups
FG_mulTable:=proc(G::list,$)
local M,i,j;
unitcellon();
M:=Matrix(nops(G),(i,j)->FG_member(FG_mul(G[i],G[j]),G),datatype=integer);
for i from 1 to nops(G) do for j from 1 to nops(G) do if (M[i,j]=0) then WARNING("Elements %1 and %2 are not multiplied",i,j) end end end;
unitcellback();
M
end:

#hfl: Groups
FG_invTable:=proc(G::list,$)
local V,i;
unitcellon();
V:=Vector(nops(G),i->FG_member(FG_inv(G[i]),G),datatype=integer);
for i from 1 to nops(G) do if (V[i]=0) then WARNING("Element %1 is not inverted",i) end end;
unitcellback();
V
end:

#hfl: Groups
FG_mulinvTable:=proc(G::list,$)
local ff,id,M,V;
if type(G[1],list) then ff:=g->map(convert,g,list) else ff:=g->convert(g,list) end;
unitcellon();
id:=table([seq(ff(FGS(G[i]))=i,i=1..nops(G))]);
M:=Matrix(nops(G),(i,j)->id[ff(FG_mul(G[i],G[j]))],datatype=integer);
V:=Vector(nops(G),i->id[ff(FG_inv(G[i]))],datatype=integer);
unitcellback();
M,V
end:

#hfl: Groups
FG_cyclic:=proc(g,n::posint,$)
local S,i;
unitcellon();
S:=Vector(n);
S[1]:=FG_unit(g);
if (n>1) then S[2]:=g end;
for i from 3 to n do S[i]:=FG_mul(g,S[i-1]) end;
unitcellback();
convert(S,list)
end:

#hfl: Groups
FG_orbitC:=proc(G::list,h,$) local g; FG_set([seq(FG_mul(FG_mul(g,h),FG_inv(g)),g=G)]) end:

#hfl: Groups
FG_factor:=proc(G::list,H::list,$) # G=FxH
local E,n,G1,GH,F1,F,i,f,h;
E:=FG_unit(G[1]);
n:=nops(G)/nops(H);
G1:=G;
GH:=table();
GH[1]:=H;
for i from 1 to n-1 do
 G1:=remove(g->evalb(FG_member(g,GH[i])>0),G1);
 GH[i+1]:=[seq(FG_mul(G1[1],h),h=H)] end;
F:=[E,GH[2][1]];
GH:=[seq(GH[i],i=3..n)];
while (nops(F)<n) do
 F1:=FG_group(F,nops(G));
 F1:=select(g->evalb(FG_member(g,G)>0),F1);
 F1:=remove(g->evalb(FG_member(g,F)>0),F1);
 for f in F1 do GH:=remove(G->evalb(FG_member(f,G)>0),GH) end;
 if (nops(GH)=0)
  then F:=[op(F),op(F1)]; break
  else F:=[op(F),op(F1),GH[1][1]]; GH:=subsop(1=NULL,GH) end end;
F
end:

#hfl: Groups
FG_testInvSG:=proc(S::list,G::list,$)
local b,h;
if FG_in(S,G) then
 b:=true;
 for h in S do if not(FG_in(FG_orbitC(G,h),S)) then b:=false; break end end;
 b
else false end
end:

#hfl: Groups
FG_testSDP:=proc(G::list,G1::list,G0::list,$)
local E,b,g1;
E:=FG_unit(G[1]);
b:=true;
for g1 in G1 do if not(FG_ident(g1,E)) then
 if (FG_member(g1,G0)>0) then b:=false; break end end end;
b and FG_equal(G,FG_prod(G1,G0)) and FG_in(G1,G) and FG_testInvSG(G0,G)
end:

#hfl: Groups
FG_testDP:=(G::list,G1::list,G0::list)->evalb(FG_testSDP(G,G1,G0) and FG_testInvSG(G1,G)):





################################################################################
#cat: Representations of groups

#hfl: FG_orbit
FG_orbit:=proc(G::list,p::Vector,output::string:="p",origin2::list:=origin,iszero::procedure:=(x->evalb(FG_simplify(x)=0)),$)
local ident,v,M,out,key,V,i,j;
unitcellon();
ident:=(p1,p2)->foldl(`and`,seq(iszero(v),v=p1-p2));
M:=Matrix(map(v->map2(op,1,v),[ListTools[Categorize]((u,v)->ident(u[2],v[2]),[seq([i,FG_app(G[i],p,origin2)],i=1..nops(G))])]),datatype=integer);
out:=NULL;
for key in output do
 if   (key="M") then out:=out,M
 elif (key="i") then out:=out,[seq(i,i=M[..,1])]
 elif (key="p") then out:=out,[seq(FG_app(G[i],p,origin2),i=M[..,1])]
 elif (key="g") then out:=out,[seq([i,seq(v,v=FG_appt(G[i],p,origin2))],i=M[..,1])]
 elif (key="h") then out:=out,[seq([j,seq(v,v=FG_appt(G[j],p,origin2))],j=M[1,..])]
 elif (key="V") then
  V:=Vector(nops(G),datatype=integer);
  for i from 1 to Dim2(M)[1] do for j from 1 to Dim2(M)[2] do V[M[i,j]]:=i end end;
  out:=out,V
 else error("Unrecognized key=%1 in output=%2",key,output) end end;
unitcellback();
out
end:


#hfl: FG_RegRep
FG_RegRep:=proc(G::list,p0::{undefined,Vector,list(Vector)}:=undefined,iszero::procedure:=(x->evalb(FG_simplify(x)=0)),$)
local ident,n,T,i,j,k,A,g,g1,P,m,p;
unitcellon();
ident:=(p1,p2)->foldl(`and`,seq(iszero(v),v=p1-p2));
n:=nops(G);
T:=Vector(n);
if (p0=undefined) then
 for i from 1 to n do
  g:=G[i];
  A:=Matrix(n,datatype=integer);
  for j from 1 to n do
   g1:=FG_mul(g,G[j]);
   for k from 1 to n while not(FG_ident(G[k],g1)) do end;
   A[k,j]:=1 end;
  T[i]:=Matrix(A) end
else
 P:=`if`(type(p0,list),p0,FG_orbit(G,p0,"p"));
 m:=nops(P);
 for i from 1 to n do
  g:=G[i];
  A:=Matrix(m,datatype=integer);
  for j from 1 to m do
   p:=FG_app(g,P[j]);
   for k from 1 to m while not(ident(P[k],p)) do end;
   A[k,j]:=1 end;
  T[i]:=Matrix(A) end end;
unitcellback();
convert(T,list)
end:



################################################################################
#cat: Representations of predefined groups
#hfl: FGR
#toc: _Overview
#Representations of predefined groups

#hfl: FGR
FGR_ChTb:=proc(G::record,{fulloutput::boolean:=false},$)
local n,M,i,gen;
if fulloutput then
 n:=nops(G:-NC);
 M:=Matrix(n+1):
 M[1,1]:=G:-name;
 for i from 1 to n do M[1,i+1]:=cat(`if`(G:-NC[i]=1,"",convert(G:-NC[i],string)),G:-nameC[i]) end;
 M[2..n+1,1]:=Vector(G:-nameR);
 M[2..n+1,2..n+1]:=Matrix([seq(map(FG_simplify@LinearAlgebra[Trace],G:-GFC(gen)),gen=G:-genR)]);
 M
else
 Matrix([seq(map(FG_simplify@LinearAlgebra[Trace],G:-GFC(gen)),gen=G:-genR)]) end
end:

#hfl: FGR
FGR_RepU:=proc(R0::list,R::list,$)
local N,n,d,P,g,lsV;
N:=nops(R);
n,d:=[op(R0[1])][1],[op(R[1])][1];
P:=Matrix(n):
for g from 1 to N do P:=FGS(P+conjugate(R[g][1,1])*R0[g]) end;
lsV:=map(v->FGS(LinearAlgebra[Normalize](FGS(v),2)),LinearAlgebra[GramSchmidt](LinearAlgebra[ColumnSpace](P)));
map(v->Matrix(n,d,(i,alpha)->FG_simplify(d/N*add(conjugate(R[g][alpha,1])*(R0[g].v)[i],g=1..N))),lsV)
end:

#hfl: FGR
FGR_RepU2:=proc(R0::list,R::list,nu::nonnegint,$)  # Is not efficient
local N,n,d,P,P1,g,lsV,j;
if (nu=0) then return [] end;
N:=nops(R);
n,d:=[op(R0[1])][1],[op(R[1])][1];
j:=nu;  P:=Matrix(n,nu):
for g from 1 to N do P:=P+conjugate(R[g][1,1])*R0[g][1..-1,1..j] end:
while (LinearAlgebra[Rank](P)<nu) do
 j:=j+1:  P1:=Vector(n):
 for g from 1 to N do P1:=P1+conjugate(R[g][1,1])*R0[g][1..-1,j] end:
 P:=Matrix([P,P1])  end;
lsV:=map(FGS,LinearAlgebra[GramSchmidt](LinearAlgebra[ColumnSpace](P),'normalized'));
map(v->Matrix(n,d,(i,alpha)->FG_simplify(d/N*add(conjugate(R[g][alpha,1])*(R0[g].v)[i],g=1..N))),lsV)
end:

#hfl: FGR
FGR_RepDec:=proc(R::list,G::record,{aslist::boolean:=false},$)
local tb,m,lsU,d,n,U;
tb:=table():
for m from 1 to nops(G:-nameR) do
 lsU:=FGR_RepU(R,G:-GF(G:-genR[m]));
 if (nops(lsU)>0) then tb[m]:=`if`(aslist,[G:-nameR[m],lsU],[G:-nameR[m],Matrix(lsU),Dim2(lsU[1])[2]]) end end;
#  if aslist then
#   tb[m]:=[G:-nameR[m],lsU]
#  else
#   d:=Dim2(lsU[1])[2];
#   n:=nops(lsU);
#   U:=Matrix(Dim2(lsU[1])[1],d*n);
#   for i from 1 to n do U[..,d*(i-1)+1..d*i]:=lsU[i] end;
#   tb[m]:=[G:-nameR[m],U,d] end end end;
convert(tb,list)
end:

#hfl: FGR
FGR_Ch:=(R::list,G::record)->[seq(FG_simplify(LinearAlgebra[Trace](R[i])),i=FGR_indexC(G))]:

#hfl: FGR
FGR_RepDecCh:=(chi::list,ChTb::Matrix,NC::list)->map(FG_simplify,convert((1/add(v,v=NC)).map(conjugate,ChTb).Vector(zip(`*`,NC,chi)),list)):

#hfl: FGR
FGR_indexC:=(G::record)->map(FG_member,G:-GFC(G:-gen),G:-GF(G:-gen)):

#hfl: FGR
FGR_dimR:=(G::record)->map(v->op(1,v[1])[1],G:-genR):





################################################################################
#cat: Sites and dimers
#hfl: FGS
#toc: _Overview
#Sites and dimers

#hfl: FGS
FGS_init:=proc( SG::{string,list}, P::{Matrix,list(Vector)}, SS::set:={}, origin2::list:=[], { maxdev::numeric:=1e-3, notable::boolean:=false, printout::boolean:=false,
 pcoo::procedure:=(x->sprintf(`if`(type(x,integer),"%d ",`if`(type(x,rational),"%a",`if`(type(x,realcons),"%.3f","%a"))),x)), wcoo::nonnegint:=0 },$)
local v,l,G,s,stab1,no,o,nm,lsm,oi2s,P1,m1,orb1,lsi,lsj,i2s,ns1,ls,dev,mis,iperm,i,geninv,k,r, x,m,t,u;
v:=indets(P);  if (SS={}) then sites_SS:={seq(v[l]=parvalues[l],l=1..nops(v))} else if (indets(SS)=v) then sites_SS:=SS else error("Inconsistent indets(P)=%1 and SS=%2",v,SS) end end;
sites_floor:=`if`(sites_SS={},floor,x->floor(eval(x,sites_SS)));
floor9:=op(FG_floor);
FG_floor:=op(sites_floor);
sites_pcoo:=pcoo;
sites_wcoo:=`if`(wcoo>0,wcoo,1+max(map(length@sites_pcoo,`if`(type(P,Matrix),[1/3,seq(v,v=P)],[1/3,seq(seq(u,u=v),v=P)]))));
G:=`if`(type(SG,list),SG,SymmetryGroup(SG,origin2));
sites_SG:=`if`(type(SG,list),"custom",SG);
sites_G:=G;
sites_ng:=nops(G);
sites_affine:=type(G[1],list);
sites_nd:=Dim2(`if`(sites_affine,G[1][1],G[1]))[2];
if type(SG,list) then
 if (origin2=[]) then origin:=[`if`(sites_affine,0,onpd)$sites_nd] else if (nops(origin2)<>sites_nd) then error("Inconsistent sites_nd=%1 and origin2=%2",sites_nd,origin2) else origin:=origin2 end end;
 FG_GroupAlgebra(`if`(sites_affine,"affine","matrix")) end;
sites_setting:=setting;
sites_origin:=origin;
sites_npd:=map(evalb@`=`,origin,onpd);
sites_id:=table([seq(`if`(sites_affine,map(convert,G[i],list),convert(G[i],list))=i,i=1..sites_ng)]);
# main
sites_orb,sites_gen,sites_oi2s,stab1:=table(),table(),table(),table();
if type(P,list) then # input 1: only orbit roots are given
 no:=nops(P);
 for o from 1 to no do
  sites_orb[o],sites_gen[o],stab1[o],sites_oi2s[o]:=FG_orbit(G,P[o],"pghV",`if`(hastype(P[o],float),(x->evalf(abs(x))<maxdev),NULL));
  if printout then printf("(%{c,}*s) %3d %3d\n",sites_wcoo,map(sites_pcoo,sites_orb[o][1]),nops(sites_gen[o]),nops(stab1[o])) end end;
 sites_P:=Matrix([seq(op(sites_orb[o]),o=1..no)]);
 sites_m2os:=[seq(seq([o,s],s=1..nops(sites_gen[o])),o=1..no)]
else # input 2: all sites positions are given
 nm:=Dim2(P)[2];
 lsm,sites_m2os,oi2s:=[$1..nm],Vector(nm),table();
 P1:=Matrix([seq(FG_app(G[1],P[..,m],true),m=1..nm)]);
 for o from 1 to infinity do
  m1:=lsm[1];
  orb1,lsi,lsj,i2s:=FG_orbit(G,P1[..,m1],"pihV",`if`(hastype(P1[..,m1],float),(x->evalf(abs(x))<maxdev),NULL));  lsj:=map2(op,1,lsj);  ns1:=nops(orb1);
  if printout then printf("(%{c,}*s) %3d %3d\n",sites_wcoo,map(sites_pcoo,P[..,m1]),ns1,nops(lsj)) end;
  if (nops(lsm)<ns1) then error("Not enough points (%1<%2) for P[%3]=%4",nops(lsm),ns1,m1,convert(P[..,m1],list)) end;
  ls,v,dev:=IdentifyPairs(lsm[2..],[$2..ns1],(m,s)->max(seq(abs(evalf(eval(v,sites_SS))),v=P1[..,m]-orb1[s])),"vrm",'nolist');
  if (dev>maxdev) then print(ls); error("Max deviation exceeded (%1>%2) for P[%3]=%4",dev,maxdev,m1,convert(P[..,m1],list)) end;
  lsm:=lsm[2..][v];
  mis:=Sort([[m1,1,1],seq([v[1],lsi[v[2]],v[2]],v=ls)],[1]); # monomer, group element, old site index
  iperm:=SortIdx(mis,[3],'nolist');  for i from 1 to sites_ng do i2s[i]:=iperm[i2s[i]] end;  sites_oi2s[o]:=i2s; # reorder old site indexes by monomers
  for s from 1 to ns1 do sites_m2os[mis[s][1]]:=[o,s] end;
  sites_orb[o]:=[seq(P[..,v[1]],v=mis)];
  sites_gen[o]:=[seq([mis[s][2],seq(round(t),t=P[..,mis[s][1]]-FG_app(G[mis[s][2]],P[..,m1]))],s=1..ns1)];
  stab1[o]:=[seq([i,seq(round(t),t=P[..,m1]-FG_app(G[i],P[..,m1]))],i=lsj)];
  if (lsm=[]) then no:=o; break end end;
 sites_P:=copy(P) end;
# initialize rest of internal varibles
sites_gen:=convert(sites_gen,list);
sites_stab1:=convert(stab1,list);
sites_no:=nops(sites_gen);
sites_ns:=map(nops,sites_gen);
sites_orb:=convert(sites_orb,list);
sites_P0:=map2(op,1,sites_orb);
sites_oi2s:=convert(sites_oi2s,list);
sites_m2os:=convert(sites_m2os,list);
sites_nm:=nops(sites_m2os);
v:=table([seq(op(sites_m2os[m])=m,m=1..sites_nm)]);
sites_os2m:=[seq([seq(v[o,s],s=1..sites_ns[o])],o=1..sites_no)];
sites_t0:=Matrix([seq(FG_appt(FG_unit(sites_G[1]),sites_P[..,m]),m=1..sites_nm)],datatype=integer);
if (origin2=[]) then for k from 1 to sites_nd do if not(sites_npd[k]) then
 r:=[min,max](eval(sites_P[k,..],sites_SS));
 if (r[1]<sites_origin[k] or r[2]>=1+sites_origin[k]) then
  if (r[2]-r[1]>=1) then WARNING("Too large range of coordinate #%1: from %2 to %3",k,op(r)) else # this should not cause a problem in FGS commands
   sites_origin[k]:=piecewise(r[1]>=0 and r[2]<1,0,r[1]>=-1/2 and r[2]<1/2,-1/2,floor(r[1]*360)/360);
   if not(type(2*sites_origin[k],integer)) then WARNING("Nonstandard origin for coordinate #%1: %2",k,sites_origin[k]) end end end end end end;
sites_stab,sites_ois2s,sites_ois2t:=[],[],[];
sites_table:=not(notable);
if sites_table then FGS_get("ois2s,ois2t") end;
dimers_orb:=table();
dimers_id:=table();
dimers_sort:=ost->ost[2]*1000000-add(ost[2+k]*[10000,100,1][k],k=1..sites_nd); # max sites_nd is 3
NULL
end:

#hfl: FGS
FGS_get:=proc(keys::string,$) # obsolete notations np=no, ng=ns, pi2s=oi2s
local ls,tb,geninv,h,o,s,i;
ls:=StringTools[Split](keys,",");
if (member("stab",ls) and sites_stab=[]) then
 tb:=table();
 for o from 1 to sites_no do
 geninv:=map(FGS_inv,sites_gen[o]);
 tb[o]:=[sites_stab1[o],seq(map(h->FGS_mul(FGS_mul(sites_gen[o][s],h),geninv[s]),sites_stab1[o]),s=2..sites_ns[o])] end;
 sites_stab:=convert(tb,list) end;
if (sites_ois2s=[] and (member("ois2s",ls) or member("ois2t",ls))) then
 sites_ois2s:=[seq(Matrix(sites_ng,sites_ns[o],(i,s)->sites_oi2s[o][FGS_g2it(FG_mul(sites_G[i],sites_G[sites_gen[o][s][1]]))[1]],datatype=integer),o=1..sites_no)] end;
if (sites_ois2t=[] and member("ois2t",ls)) then
 tb:=table([seq(o=Array(1..sites_ng,1..sites_ns[o],1..sites_nd,datatype=integer),o=1..sites_no)]);
 for o from 1 to sites_no do for i from 1 to sites_ng do for s from 1 to sites_ns[o] do tb[o][i,s,..]:=sites_t0[..,sites_os2m[o,sites_ois2s[o][i,s]]]-FG_appt(sites_G[i],sites_orb[o][s]) end end end;
 sites_ois2t:=convert(tb,list) end;
if (sites_ois2s<>[] and sites_ois2t<>[]) then sites_table:=true end;
seq(
`if`(s="SS",sites_SS,
`if`(s="pcoo",sites_pcoo,
`if`(s="wcoo",sites_wcoo,
`if`(s="SG",sites_SG,
`if`(s="G",sites_G,
`if`(s="ng",sites_ng,
`if`(s="nd",sites_nd,
`if`(s="npd",sites_npd,
`if`(s="origin",sites_origin,
`if`(s="id",op(sites_id),
`if`(s="P",sites_P,
`if`(s="t0",sites_t0,
`if`(s="P0",sites_P0,
`if`(s="no",sites_no,
`if`(s="orb",sites_orb,
`if`(s="gen",sites_gen,
`if`(s="stab1",sites_stab1,
`if`(s="stab",sites_stab,
`if`(s="ns",sites_ns,
`if`(s="nm",sites_nm,
`if`(s="m2os",sites_m2os,
`if`(s="os2m",sites_os2m,
`if`(s="oi2s",sites_oi2s,
`if`(s="ois2s",sites_ois2s,
`if`(s="ois2t",sites_ois2t,
undefined))))))))))))))))))))))))),s=ls)
end:

#hfl: FGS
FGS_correctit:=proc(it::list,$) local k; [it[1],seq(`if`(sites_npd[k],0,it[1+k]),k=1..sites_nd)] end:

#hfl: FGS
FGS_correctost:=proc(ost::list,$) local k; [op(..2,ost),seq(`if`(sites_npd[k],0,ost[2+k]),k=1..sites_nd)] end:

#hfl: FGS
FGS_it2g:=proc(it,$) local g;  g:=sites_G[it[1]];  `if`(sites_affine,[g[1],g[2]+Vector(sites_nd,it[2..])],g) end:

#hfl: FGS
FGS_g2it:=proc(g,$) local tg;  if sites_affine then  tg:=map(reduce0,g[2]);  [sites_id[map(convert,[g[1],tg],list)],seq(v,v=(g[2]-tg))] else [sites_id[convert(g,list)],0$sites_nd] end end:

#hfl: FGS
FGS_inv:=it->FGS_g2it(FG_inv(FGS_it2g(it))):

#hfl: FGS
FGS_mul:=(it1,it2)->FGS_g2it(FG_mul(FGS_it2g(it1),FGS_it2g(it2))):

#hfl: FGS
FGS_s2it:=ost->sites_gen[ost[1]][ost[2]]+[0,op(3..,ost)]:

#hfl: FGS
FGS_app:=proc(it,ost,$)
local o,s,i,g,k2,it2,it3,t;
if sites_table then
 o,s,i:=ost[1],ost[2],it[1];
 g:=sites_G[i];  if type(g,list) then g:=g[1] end;
 [o,sites_ois2s[o][i,s],seq(sites_ois2t[o][i,s,k]+it[1+k]+add(g[k,k2]*ost[2+k2],k2=1..sites_nd),k=1..sites_nd)]
else
 unitcelloff();
 o:=ost[1];
 it2:=FGS_mul(it,FGS_s2it(ost));
 s:=sites_oi2s[o][it2[1]];
 it3:=FGS_s2it([o,s,0$sites_nd]);
 t:=(FG_app(FGS_it2g(it2),sites_P0[o])-FG_app(FGS_it2g(it3),sites_P0[o]));
 unitcellback();
 [o,s,seq(round(v),v=t)] end
end:

#hfl: FGS
FGS_s2m:=proc(s::list,$)
if    type(s,listlist)    then map(FGS_s2m,s)
elif (nops(s)=2         ) then  sites_os2m[s[1]][s[2]]
elif (nops(s)=sites_nd+2) then [sites_os2m[s[1]][s[2]],op(3..,s)]
elif (nops(s)=sites_nd+3) then [sites_os2m[s[1]][1]   ,sites_os2m[s[2]][s[3]],op(4..,s)]
elif (nops(s)=sites_nd+4) then [sites_os2m[s[2]][s[1]],sites_os2m[s[3]][s[4]],op(5..,s)]
else error("Unrecognized input: %1",s) end
end:

#hfl: FGS
FGS_m2s:=proc(m::{posint,list},$)
if    type(m,listlist)    then map(FGS_m2s,m)
elif  type(m,posint)      then  sites_m2os[m]
elif (nops(m)=sites_nd+1) then [op(sites_m2os[m[1]]),op(2..,m)]
elif (nops(m)=sites_nd+2) then [op(sites_m2os[m[1]][[2,1]]),op(sites_m2os[m[2]]),op(3..,m)]
else error("Unrecognized input: %1",m) end
end:

#hfl: FGS
FGS_p2s:=proc(p::Vector,maxdev::numeric:=0,SS::set:=sites_SS,{noerror::boolean:=false},$)
local frac2,ost,o,s,t, v,k;
frac2:=v->v-round(v);
ost:=[];
for o from 1 to sites_no while (ost=[]) do for s from 1 to sites_ns[o] do
 t:=FGS(eval(p-sites_orb[o][s],SS));
 if (sites_affine and foldl(`and`,seq(abs(`if`(sites_npd[k],t[k],frac2(t[k])))<=maxdev,k=1..sites_nd)) or foldl(`and`,seq(abs(v)<=maxdev,v=t))) then ost:=[o,s,seq(round(v),v=t)]; break end end end;
if (noerror or ost<>[]) then ost else error("Cannot identify site for p=%1",p) end
end:

#hfl: FGS
FGS_s2p:=proc(s::list,$)  # old vesrion  ost->FG_app(FGS_it2g(FGS_s2it(ost)),sites_P0[ost[1]],false)
if    type(s,[list,list]) then FGS_s2p(s[2])-FGS_s2p(s[1])
elif (nops(s)=2         ) then sites_orb[s[1]][s[2]]
elif (nops(s)=sites_nd+2) then sites_orb[s[1]][s[2]]+Vector(s[3..])
elif (nops(s)=sites_nd+3) then sites_orb[s[2]][s[3]]+Vector(s[4..])-sites_orb[s[1]][1]
elif (nops(s)=sites_nd+4) then sites_orb[s[3]][s[4]]+Vector(s[5..])-sites_orb[s[2]][s[1]]
else error("Unrecognized input: %1",s) end
end:

#hfl: FGS
FGS_m2p:=proc(m::{posint,list},$)  # old vesrion  mt->sites_P[..,mt[1]]+`if`(sites_affine,Vector(mt[2..]),NULL)
if    type(m,[list,list]) then FGS_m2p(m[2])-FGS_m2p(m[1])
elif  type(m,posint)      then sites_P[..,m]
elif (nops(m)=sites_nd+1) then sites_P[..,m[1]]+Vector(m[2..])
elif (nops(m)=sites_nd+2) then sites_P[..,m[2]]+Vector(m[3..])-sites_P[..,m[1]]
else error("Unrecognized input: %1",m) end
end:

#hfl: FGS
FGS_spick:=proc(o::posint,pickf::procedure,dev::numeric:=1e-7,$)
local V,s,V2;
V:=[seq(pickf(eval(sites_orb[o][s],sites_SS)),s=1..sites_ns[o])];
s:=op(MaxIdx(V));
V2:=select(`>`,V,V[s]-dev);
if (nops(V2)>1) then error("Change pickf, because of multiple maxima: %1",V2) end;
s
end:

#hfl: FGS
FGS_d2oost:=dimer->[dimer[1][1],op(FGS_app(FGS_inv(FGS_s2it(dimer[1])),dimer[2]))]:

#hfl: FGS
FGS_d2oost1:=proc(dimer,$)
local oost,oost1,o,ost,osts,p,V,k,V2, it;
oost:=FGS_d2oost(dimer);
if assigned('dimers_id[op(oost)]') then oost1:=dimers_id[op(oost)] else
 o,ost:=oost[1],oost[2..];
 osts:=convert({seq(FGS_app(it,FGS_app(sites_gen[o][1],ost)),it=sites_stab1[o])},list);
 osts:=Sort(osts,dimers_sort);
 oost1:=[o,op(osts[1])];
 dimers_orb[op(oost1)]:=osts;
 for ost in osts do dimers_id[o,op(ost)]:=oost1 end end;
oost1
end:

#hfl: FGS
FGS_d2oost2:=proc(dimer,$)
local oost1,oost2,ls,p,V,k,V2, v;
oost1:=FGS_d2oost1(dimer);
oost2:=FGS_d2oost1(dimer[[2,1]]);
if (oost1<>oost2) then oost1:=`if`(oost1[1]=oost2[1],MinVal([oost1,oost2],v->dimers_sort(v[2..])),MinVal([oost1,oost2],[1])) end;
oost1
end:

#hfl: FGS
FGS_dorb:=proc(dimer::list,$)
local oost;
if type(dimer,listlist) then oost:=FGS_d2oost1(dimer)
elif (nops(dimer)=4+sites_nd) then oost:=FGS_d2oost1([[dimer[2],dimer[1],0$sites_nd],dimer[3..]])
elif (nops(dimer)=3+sites_nd) then oost:=FGS_d2oost1([[dimer[1],1,0$sites_nd],dimer[2..]])
else error("Unrecognized argument dimer::{[ost$2],soost,oost} but received %1",dimer) end;
oost,dimers_orb[op(oost)]
end:

#hfl: FGS
FGS_dorb2:=proc(dimer::list,$)
local oost,dorb,oost2,dorb2;
oost:=FGS_dorb(dimer)[1];
oost:=FGS_d2oost2([[oost[1],1,0$sites_nd],oost[2..]]);
oost,dorb:=FGS_dorb(oost);
oost2,dorb2:=FGS_dorb([oost[2..],[oost[1],1,0$sites_nd]]);
oost,`if`(oost=oost2,dorb,[op(dorb),op(dorb2)]);
end:

#hfl: FGS
FGS_dpick:=proc(oost::list(integer),pickf::procedure,s1s::list(posint):=[1$sites_no],dev::numeric:=1e-7,{undirected::boolean:=false},$)
local o1,o2,s1,s2,oost1,oost2,soosts,ost,V,soost,k,V2,v;
o1,o2:=oost[1],oost[2];
s1,s2:=s1s[o1],s1s[o2];
oost1:=FGS_d2oost1([[o1,1,0$sites_nd],oost[2..]]);
if undirected then
 oost2:=FGS_d2oost1([oost[2..],[o1,1,0$sites_nd]]);
 if not(oost=oost1 or oost=oost2) then error("Unrecognized oost=%1, should be one of %2, %3",oost,oost1,oost2) end
else
 if not(oost=oost1) then error("Unrecognized oost=%1, should be %2",oost,oost1) end end;
soosts:=[seq([s1,o1,op(FGS_app(sites_gen[o1][s1],ost))],ost=dimers_orb[op(oost1)])];
if (undirected and oost1<>oost2) then soosts:=[op(soosts),seq([s2,o2,op(FGS_app(sites_gen[o2][s2],ost))],ost=dimers_orb[op(oost2)])] end;
V:=map(soost->pickf(eval(sites_orb[soost[2]][soost[1]],sites_SS),eval(FGS_s2p(soost[3..]),sites_SS)),soosts);
k:=op(MaxIdx(V));
V2:=select(`>`,V,V[k]-dev);
if (nops(V2)>1) then
 map(v->printf("[%{c,}d] (%{c,}*s) (%{c,}*s) %a\n",Vector(v[1]),sites_wcoo,map(sites_pcoo,v[2]),sites_wcoo,map(sites_pcoo,v[3]),v[4]),
   select(v->evalb(v[-1]>V[k]-dev),map(soost->[soost,sites_orb[soost[2]][soost[1]],FGS_s2p(soost[3..]),pickf(eval(sites_orb[soost[2]][soost[1]],sites_SS),eval(FGS_s2p(soost[3..]),sites_SS))],soosts)));
 error("Change pickf, because of multiple maxima: %1",V2) end;
soosts[k]
end:

#hfl: FGS
FGS_dimersR:=proc(o1::posint,o2::posint,R::realcons,M::Matrix,{printout::{boolean,nonnegint}:=false},$)
local ncol,MM,pMMp,p,k,k2,a,v,ia,p0,isfloat,delta,R2,id,N,ncmax,s2,dp,nc,iterator,tt,oosts,u,ls;
if not(sites_nd=3 and [Dim2(M)]=[3,3]) then error("Current implementation of 'ia' is limited to 3D space") end;
ncol:=`if`(printout=false,-1,`if`(printout=true,1,printout));
MM:=map(FG_simplify,LinearAlgebra[Transpose](M).M);
pMMp:=p->add(MM[k,k]*p[k]^2+2*add(MM[k,k2]*p[k]*p[k2],k2=k+1..sites_nd),k=1..sites_nd);
a:=[seq(sqrt(add(v^2,v=M[..,k])),k=1..sites_nd)];
ia:=map2(`/`,a[1]*a[2]*a[3]/LinearAlgebra[Determinant](M),a); # works only in 3D space
p0:=sites_orb[o1][1];
isfloat:=not(foldl(`and`,seq(seq(type(eval(v,sites_SS),rational),v=u),u=[MM,p0,sites_orb[o2][1]])));
delta:=`if`(isfloat,1e-4,0);
R2:=simplify((R+delta)^2);  if isfloat then R2:=evalf(R2) else if not(type(R2,numeric)) then error("Non-numeric R2=%1",R2) end end;
id,N,ncmax:=table(),0,[[0,0]$sites_nd];
for s2 from 1 to FGS_get("ns")[o2] do
 dp:=eval(FGS_s2p([o2,s2,0$sites_nd])-p0,sites_SS);
 nc:=[seq(`if`(sites_npd[k],[0,0],[floor(-dp[k]-R*ia[k]-delta),ceil(-dp[k]+R*ia[k]+delta)]),k=1..sites_nd)];
 ncmax:=[seq([min(ncmax[k],nc[k]),max(ncmax[k],nc[k])],k=1..sites_nd)];
 N:=N+mul(v[2]-v[1]+1,v=nc);
 iterator:=combinat[cartprod](map(v->[$v[1]..v[2]],nc));
 while not(iterator[finished]) do
  tt:=iterator[nextvalue]();
  if (pMMp(dp+Vector(tt))<=R2) then id[o1,o2,s2,op(tt)]:=FGS_d2oost1([[o1,1,0$sites_nd],[o2,s2,op(tt)]]) end end end;
oosts:=Sort(convert({entries(id,nolist)},list),v->add(u^2,u=M.eval(FGS_s2p(v[2..])-p0,sites_SS)));
if (ncol>=0) then printf("|oosts|=%d, |dimers|=%d, ball/supercell=%.2f, nc=[%{c,}s], |stab|=%d, p1=(%{c,}s)\n",
 nops(oosts), nops([indices(id)]), nops([indices(id)])/N, Vector(map(v->sprintf("%d..%d",op(v)),ncmax)), nops(sites_stab1[o1]), map(TrimRight@sites_pcoo,sites_orb[o1][1])) end;
if (ncol>=1) then
 ls:=map(v->[v,nops(FGS_dorb(v)[2]),add(u^2,u=M.eval(FGS_s2p(v[2..])-p0,sites_SS))],oosts);
 PrintColumns(map(v->sprintf("[%d,%d,%2d,%2d,%2d,%2d] (%{c,}*s) %3d %s",op(v[1]),sites_wcoo,map(sites_pcoo,FGS_s2p(v[1][2..])),v[2],sprintf(`if`(isfloat,"%6.3f","%a"),v[3])),ls),ncol) end;
oosts
end:

#hfl: FGS
FGS_dimersR2:=proc(R::realcons,M::Matrix,mmts::list(list):=[],{undirected::boolean:=false,printout::{boolean,nonnegint}:=false},$)
local dimers,v,tb,mmtid,o1,o2,oosts,oost,id,osts,s1,it1,m1,ost;
dimers:=[seq([[v[2],v[1],0$sites_nd],v[3..]],v=map(FGS_m2s,mmts))];
if undirected then dimers:=[op(dimers),seq(v[[2,1]],v=dimers)] end;
tb:=Classify2({seq(FGS_d2oost1(v),v=dimers)},[..2]);
mmtid:=table();
for o1 from 1 to sites_no do for o2 from 1 to sites_no do # use of o1-o2 symmetry gives minor increase in efficiency but large increase in complexity
 oosts:=FGS_dimersR(o1,o2,R,M,':-printout'=printout);
 oosts:={op(oosts)} union `if`(assigned('tb[[o1,o2]]'),tb[[o1,o2]],{});
 for oost in oosts do
  id:=`if`(undirected,FGS_d2oost2,FGS_d2oost1)([[o1,1,0$sites_nd],oost[2..]]);
  osts:=FGS_dorb(oost)[2];
  for s1 from 1 to sites_ns[o1] do
   it1:=sites_gen[o1][s1];
   m1:=sites_os2m[o1,s1];
   for ost in osts do mmtid[m1,op(FGS_s2m(FGS_app(it1,ost)))]:=id end end end end end;
if (printout<>false) then printf("|oosts|=%d, |dimers|=%d\n",nops({entries(mmtid)}),nops([indices(mmtid)])) end;
op(mmtid)
end:

#hfl: FGS
Encodesoost:=proc(soost::list(integer),{short::boolean:=false},$)
local s1,o1,o,st,q1,q,v;
s1,o1,o,st:=op(..3,soost),soost[4..];
if short then if (o1<>o) then error("Option 'short' is inconsistent with given soost=%1",soost) else q1,q:="","" end else q1,q:=StringTools[Char](64+o1),StringTools[Char](64+o) end;
if (o1=o and s1=st[1] and foldl(`and`,seq(evalb(v=0),v=st[2..]))) then sprintf("e%s",q1) else sprintf("t%s%s%s",q1,q,MolMod[EncodeTranslation](st)) end
end:

#hfl: FGS
Decodesoost:=proc(s::string,nd::nonnegint,s1s::{posint,list(posint)}:=1,$)
local o1,s1,o,r,st;
if (s[1]="e") then
 if (length(s)=1) then o1:=1 elif (length(s)=2 and StringTools[IsUpper](s[2])) then o1:=StringTools[Ord](s[2])-64 else error("Unrecognized code: %1",s) end;
 s1:=`if`(type(s1s,posint),s1s,s1s[o1]);
 [s1,o1,o1,s1,0$sites_nd]
elif (s[1]="t") then
 if StringTools[IsUpper](s[2..3]) then o1,o,r:=StringTools[Ord](s[2])-64,StringTools[Ord](s[3])-64,s[4..] else o1,o,r:=1,1,s[2..] end;
 s1:=`if`(type(s1s,posint),s1s,s1s[o1]);
 st:=op(MolMod[DecodeTranslation](r));
 [s1,o1,o,op(st)]
else error("Unrecognized prefix in %1",s) end
end:


#Obsolete code
FGS_ReduceDimer:=proc( dimer::[[posint,integer,integer,integer,posint],[posint,integer,integer,integer,posint]], dir::Vector(3):=<10000,100,1>, dev::numeric:=1e-7, $)
local p1,p2,it1,P1,P2,V,k,V2,it2,sto2;
WARNING("FGS_ReduceDimer command is obsolete");
unitcelloff();
p1,p2:=dimer[1][-1],dimer[2][-1];
it1:=FGS_inv(FGS_s2it(dimer[1]));
P1:=sites_P0[p1];
P2:=FG_app(FGS_it2g(it1),FGS_s2p(dimer[2]),false);
V:=map(it->evalf(dir.(FG_app(FGS_it2g(it),P2,false)-P1)),sites_stab1[p1]);
k:=op(MaxIdx(V));
V2:=select(`>`,V,V[k]-dev);
if (nops(V2)>1) then error("Change dir, because of multiple maxima: %1",V2) end;
it2:=sites_stab1[p1][k];
sto2:=FGS_app(FGS_mul(it2,it1),dimer[2]);
unitcellback();
[[1,0$(nops(it1)-1),p1],sto2]
end:

FGS_ReduceDimer2:=proc( dimer::[[posint,integer,integer,integer,posint],[posint,integer,integer,integer,posint]], dir::Vector(3):=<10000,100,1>, $)
local p1,p2,sto2a,sto2b;
WARNING("FGS_ReduceDimer2 command is obsolete");
p1,p2:=dimer[1][-1],dimer[2][-1];
if   (p1<p2) then FGS_ReduceDimer(dimer,dir)
elif (p1>p2) then FGS_ReduceDimer([dimer[2],dimer[1]],dir)
else
 sto2a:=FGS_ReduceDimer(dimer,dir)[2];
 sto2b:=FGS_ReduceDimer([dimer[2],dimer[1]],dir)[2];
 [[1,0$(nops(dimer[1])-2),p1],`if`(evalf(dir.(FGS_s2p(sto2a)-sites_P0[p1]))<evalf(dir.(FGS_s2p(sto2b)-sites_P0[p1])),sto2b,sto2a)] end
end:

FGS_UnfoldDimer2:=proc(xosto::{[posint,posint,posint,integer,integer,integer,posint],[posint,posint,integer,integer,integer,posint]},$)
local sosto,ls,e,dimer,o,sto,s,it;
WARNING("FGS_UnfoldDimer2 command is obsolete");
sosto:=`if`(nops(xosto)=7,xosto,[1,op(xosto)]);
ls:=[];
for e in [[1,2],[2,1]] do
 dimer:=FGS_ReduceDimer([[sosto[1],0,0,0,sosto[2]],sosto[3..]][e]);
 o,sto:=dimer[1][-1],dimer[2];
 ls:={op(ls),seq(seq([s,o,op(FGS_app(it,FGS_app(sites_gen[o][s],sto)))],it=sites_stab[o][s]),s=1..sites_ns[o])} end;
[op(ls)]
end:





################################################################################
#cat: Tables of groups
#hfl: Tables
#toc: _Overview
#Tables of groups

#hfl: SymmetryGroup
PrintSymmetryElement:=proc(g::{Matrix,[Matrix,Vector]},kind::{"xyz","param"}:="xyz",{maxwidth::nonnegint:=4},$)
local R,t,s,d,a,n,a0,v,s1,x,y,z;
if type(g,Matrix) then R,t:=g,Vector(op(1,g)[1]) else R,t:=op(g) end;
if (kind="param") then
 d,a,n:=RotationParam(g);
 a0:=Reduce2P(a/2/Pi);
 if not(type(a0,atomic)) then a0:=evalf(a0) end;
 a:=convert(a0,rational,3);
 if (abs(a-a0)>1e-4) then error("Cannot convert angle %1 to rational of 2*Pi",a0) end;
 s:=sprintf("%2d^%d [",d*denom(a),numer(a));
 for v in n do
  s1:=sprintf("%a",v);
  s:=cat(s,`if`(length(s1)>maxwidth,FormatFloat(evalf(v),2,'maxorder'=0),s1),",") end;
 s:=cat(s[..-2],"]");
 if (add(abs(v),v=t)<>0) then s:=sprintf("%s <%{c,}a>",s,t) end
elif (kind="xyz") then
 s:=sprintf("%{c,}a",R.<x,y,z>+t)
end;
s
end:

#hfl: SymmetryGroup
SymmetryGroup:=proc( SG0::string, T00::{undefined,Matrix,[Matrix,Vector],Vector}:=undefined, origin2::list:=[],
 { primitive::boolean:=false, output::string:="G", nosetup::boolean:=false, printout::boolean:=false},$)
local T0,cent0,SG,tag,shift,special,T1,T2a,T2b,T2,TP,prim,v,kind,cent,n,nt,fam,cell,setting2,G,T,m,nd,origin1,TR,v1,v2,k1,k2,g,out,key;
T0:=T00;  if type(T0,Vector) then T0:=[LinearAlgebra[IdentityMatrix](op(1,T0)),T0] end;
# process SG
cent0:=`if`(SearchText(SG0[1],"ABCIRFabc")>0,SG0[1],"");
SG,tag,shift,special,T1:=SG0,"","","",undefined;
if (SG="") then error("Unrecognized group (empty string)") end;
if (SG[-1]="P") then SG,prim:=SG[..-2],true else prim:=primitive end;
if (SearchText(SG[-1],"wxyz")>0) then
 SG,tag:=SG[..-2],SG[-1];
 if (SearchText(SG[-1],"wxyz")>0) then
  SG,tag:=SG[..-2],cat(SG[-1],tag) end end;
if (length(SG)>4 and SG[-4]=":" and foldl(`and`,seq(StringTools[IsDigit](v),v=SG[-3..]))) then SG,shift:=SG[..-5],SG[-3..] end;
if (length(SG)>1 and SG[-2..]=":R") then SG,special,prim:=SG[..-3],"R",false end; # old code: SG[..-3],`if`(prim,"","R")
if assigned('tbG[SG]') then
 if type(tbG[SG],string) then return SymmetryGroup(tbG[SG]) end
elif assigned('TransformationTable[SG]') then
 v:=TransformationTable[SG];
 if type(v,string) then
  SG:=v
 else
  SG:=v[1];
  T1:=`if`(nops(v)=2,v[2],v[2..3]) end
else error("Unrecognized group: %1",SG0) end;
# characterize SG
if (SG[1]="q") then
 kind,cent,n,nt,fam,cell:="rod","P",RodGroupNumber[SG],1,"a","a"
elif (SearchText(SG[1],"pabc")>0) then
 kind,cent,n,nt:="layer",StringTools[UpperCase](SG[1]),LayerGroupNumber[SG],2;
 fam:=piecewise(n<8,"m",n<49,"o",n<65,"t","h");
 cell:=fam
elif (SearchText(SG[1],"PABCIRF")>0) then
 kind,cent,n,nt:="space",SG[1],SpaceGroupNumber[SG],3;
 fam:=piecewise(n<3,"a",n<16,"m",n<75,"o",n<143,"t",n<195,"h","c");
 if (special="R" and (fam="h" or fam="c")) then cell,cent:="r","P" else cell:=fam end
else
 kind,cent,n,nt,fam,cell:="point","",0,0,"","" end;
# setup group algebra
if not(nosetup) then  setting2:=`if`(nt>0,"affine","matrix");  if (setting2<>setting) then FG_GroupAlgebra(setting2) end end;
# get G and determine transformation T
G:=`if`(special="R",SymmetryGroup(SG,':-primitive'=true,':-nosetup'=true),tbG[SG]);
if (tag="") then T2a:=undefined else if assigned('TransformationTable[tag]') then T2a:=TransformationTable[tag] else error("Unrecognized transformation tag in %1",SG0) end end;
if (shift="") then T2b:=undefined else T2b:=Vector(3,i->[0,-1/8,1/2,1/3,1/4,-1/6,1/6,-1/4,1/8,-1/3][1+op(sscanf(shift[i],"%d"))]) end; #for i from 1 to 3 do if (T1[i]>0) then T1[i]:=1/T1[i] end end;
T2:=`if`(T2b=undefined,T2a,`if`(T2a=undefined,[LinearAlgebra[IdentityMatrix](op(1,T2b)),T2b],`if`(type(T2a,list),[T2a[1],T2a[2]+T2b],[T2a,T2b])));
if (prim and cent0<>"") then
 m:=piecewise(cent0="F",4,cent0="R",3,2);
 TP:=TransformationTable[cent0];
 G:=G[1..nops(G)/m];
else
 m:=1;
 TP:=undefined end;
T:=T1;
if not(T2=undefined) then T:=`if`(T=undefined,T2,FG_mul(T,T2)) end;
if not(TP=undefined) then T:=`if`(T=undefined,TP,FG_mul(T,TP)) end;
if not(T0=undefined) then T:=`if`(T=undefined,T0,FG_mul(T,T0)) end;
# Determine origin and transform G and origin
if type(G,list) then
 nd:=Dim2(`if`(type(G[1],list),G[1][1],G[1]))[2];
 origin1:=`if`(kind="point",[1,1,1],`if`(kind="rod",[1,1,0],`if`(kind="layer",[0,0,1],`if`(kind="space",[0,0,0],[])))) end;
if (T<>undefined) then
 G:=FG_transform(G,T);
 TR:=`if`(type(T,list),T[1],T);
 nd:=Dim2(TR)[2];
 v1:=[seq(`if`(foldl(`and`,seq(evalb(origin1[k1]=0 or TR[k2,k1]=0),k1=1..nd)),0,1),k2=1..nd)];
 v2:=[seq(`if`(foldl(`and`,seq(evalb(origin1[k1]=1 or TR[k2,k1]=0),k1=1..nd)),1,0),k2=1..nd)];
 if (v1=v2) then origin1:=[seq(`if`(foldl(`and`,seq(evalb(origin1[k1]=0 or TR[k1,k2]=0),k1=1..nd)),0,1),k2=1..nd)]
            else WARNING("Origin is left unchanged because transformations are inconsistent with translations: origin1=%1, %1<>%2, TR=%2",origin,v1,v2,TR) end end;
# set origin
if not(nosetup) then
 origin1:=map(v->`if`(v=1,onpd,0),origin1);
 if (origin2=[]) then origin:=origin1 else if (map(v->`if`(v=onpd,onpd,0),origin2)=origin1) then origin:=origin2 else error("Inconsistent origin1=%1 and origin2=%2",origin1,origin2) end end end;
# output
if printout then seq(printf("%s\n",PrintSymmetryElement(g)),g=G) end;
out:=NULL;
for key in output do
 if   (key="G") then out:=out,G
 elif (key="n") then out:=out,SG
 elif (key="T") then out:=out,T
 elif (key="P") then out:=out,TP
 elif (key="m") then out:=out,m
 elif (key="k") then out:=out,[kind,cell,cent,fam,GeometricClass[SG],ArithmeticClass[SG],n,nt];  if (SG<>SG0) then WARNING("Analysis is made not for %1 but for %2",SG0,SG) end
 else error("Unrecognized key=%1 (must be one of {GnkTm}) in output=%2",key,output) end end;
out
end:


#hfl: kPoints
kPoints:=proc(SG::string,$)
local aclass,v;
aclass:=SymmetryGroup(SG,output="k")[6];
if assigned('tbK[aclass]') then  v:=tbK[aclass];  `if`(type(v,string),tbK[v],v) else error("Not implemented for SG=%1, aclass=%2",SG,aclass) end
end:


#hfl: Tables
HM2Scho:=proc(s::string,$)
local n;
if (s="23") then "T" elif (s="m-3") then "Th" elif (s="-43m") then "Td" elif (s="432") then "O" elif (s="m-3m") then "Oh" elif (s="25") then "Y" elif (s="m-5") then "Yh" elif (s="m") then "C1h" elif (s="-6") then "C3h" elif (s="-10") then "C5h" elif (s="-122m") then "D6d" elif (s="mmm") then "D2h" elif (s="-102m") then "D5h" elif (s="mm2" or s="mm") then "C2v" else
if StringTools[IsDigit](s[1]) then n:=parse(s[1]) elif (s[1]="-" and StringTools[IsDigit](s[2])) then n:=parse(s[2]) else error("Cannot determine axis order in %1",s) end;
if   (length(s)=1) then cat("C",n)
elif (length(s)=2 and s[1]="-") then cat("C",n,"i")
elif (length(s)=3 and s[2..3]="/m" and type(n,even)) then cat("C",n,"h")
elif (length(s)=2 and s[2]="2" and type(n,odd)) then cat("D",n)
elif (length(s)=3 and s[2..3]="22" and type(n,even)) then cat("D",n)
elif (length(s)=3 and s[1]="-" and s[3]="m" and type(n,odd)) then cat("D",n,"d")
elif (length(s)=4 and s[1]="-" and s[3..4]="2m" and type(n/2,even)) then cat("D",n/2,"d")
elif (length(s)=4 and s[1]="-" and s[3..4]="2m" and type(n/2,odd )) then cat("D",n/2,"h")
elif (length(s)=5 and s[2..5]="/mmm" and type(n,even)) then cat("D",n,"h")
elif (length(s)=2 and s[2]="m" and type(n,odd)) then cat("C",n,"v")
elif (length(s)=3 and s[2..3]="mm" and type(n,even)) then cat("C",n,"v")
else error("Unrecognized HM symbol: %1",s) end
end
end:

#hfl: Tables
Scho2HM:=proc(s::string,$)
local c,n;
if (s[1]="C" or s[1]="D") then
 if StringTools[IsDigit](s[-1]) then c,n:=s[1],parse(s[2..]) else c,n:=cat(s[1],s[-1]),parse(s[2..-2]) end;
 if not(type(n,posint)) then error("Cannot determine axis order in %1",s) end;
 if   (c="C" ) then sprintf("%d",n)
 elif (c="Ci") then `if`(n=2,"m",sprintf("-%d",n))
 elif (c="Ch") then `if`(n=1,"m",`if`(type(n,odd),sprintf("-%d",2*n),sprintf("%d/m",n)))
 elif (c="D" ) then `if`(n=1,"2",`if`(type(n,odd),sprintf("%d2",n),sprintf("%d22",n)))
 elif (c="Dd") then `if`(n=1,"2/m",`if`(type(n,odd),sprintf("-%dm",n),sprintf("-%d2m",2*n)))
 elif (c="Dh") then `if`(n=1,"2/m",`if`(n=2,"mmm",`if`(type(n,odd),sprintf("-%d2m",2*n),sprintf("%d/mmm",n))))
 elif (c="Cv") then `if`(n=1,"m",`if`(n=2,"mm2",`if`(type(n,odd),sprintf("%dm",n),sprintf("%dmm",n))))
 else error("Unrecognized Scho symbol: %1",s) end
elif (s="T") then "23" elif (s="Th") then "m-3" elif (s="Td") then "-43m" elif (s="O") then "432" elif (s="Oh") then "m-3m" elif (s="Y") then "25" elif (s="Yh") then "m-5" else error("Unrecognized Scho symbol: %1",s) end
end:



########################################################################

tbG:=[

# Point groups generated in PointG.mw
"1"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
NULL],
"2"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
NULL],
"3"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
NULL],
"4"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
NULL],
"5"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
NULL],
"6"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
NULL],
"8"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
<<1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
NULL],
"-1"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
NULL],
"-3"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,-1>>,
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NULL],
"m"=[
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"-6"=[
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"4/m"=[
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"-10"=[
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<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,-1>>,
NULL],
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"8/m"=[
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NULL],
"222"=[
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NULL],
"32"=[
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NULL],
"422"=[
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<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,-1>>,
NULL],
"622"=[
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NULL],
"822"=[
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<<0,-1,0>|<-1,0,0>|<0,0,-1>>,
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NULL],
"-42m"=[
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<<-1,0,0>|<0,-1,0>|<0,0,1>>,
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<<0,-1,0>|<-1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<0,1,0>|<1,0,0>|<0,0,-1>>,
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NULL],
"-3m"=[
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<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
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<<-1,0,0>|<0,1,0>|<0,0,-1>>,
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<<1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
NULL],
"-82m"=[
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<<0,1,0>|<-1,0,0>|<0,0,1>>,
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<<-1,0,0>|<0,-1,0>|<0,0,1>>,
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<<1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
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<<0,-1,0>|<-1,0,0>|<0,0,1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,-1>>,
<<0,1,0>|<1,0,0>|<0,0,1>>,
NULL],
"-5m"=[
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<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<1/4+1/4*5^(1/2),1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<1/4+1/4*5^(1/2),-1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<-1/4*5^(1/2)+1/4,1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,-1>>,
<<1/4+1/4*5^(1/2),-1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4*5^(1/2)+1/4,1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/4*5^(1/2)+1/4,-1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<1/4+1/4*5^(1/2),1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,1>>,
NULL],
"-122m"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2*3^(1/2),-1/2,0>|<1/2,-1/2*3^(1/2),0>|<0,0,-1>>,
<<1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,-1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<1/2*3^(1/2),-1/2,0>|<1/2,1/2*3^(1/2),0>|<0,0,-1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<1/2*3^(1/2),1/2,0>|<-1/2,1/2*3^(1/2),0>|<0,0,-1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,-1>>,
<<1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<-1/2*3^(1/2),1/2,0>|<-1/2,-1/2*3^(1/2),0>|<0,0,-1>>,
<<1/2*3^(1/2),-1/2,0>|<-1/2,-1/2*3^(1/2),0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2*3^(1/2),1/2,0>|<1/2,-1/2*3^(1/2),0>|<0,0,-1>>,
<<-1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<0,1,0>|<1,0,0>|<0,0,-1>>,
<<1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1/2*3^(1/2),1/2,0>|<1/2,1/2*3^(1/2),0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2*3^(1/2),-1/2,0>|<-1/2,1/2*3^(1/2),0>|<0,0,-1>>,
<<1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<0,-1,0>|<-1,0,0>|<0,0,-1>>,
<<-1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
NULL],
"mmm"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
NULL],
"-62m"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,-1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,-1>>,
<<-1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
NULL],
"4/mmm"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<0,1,0>|<1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,-1,0>|<-1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<0,-1,0>|<1,0,0>|<0,0,-1>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,1,0>|<-1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<0,-1,0>|<-1,0,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,1,0>|<1,0,0>|<0,0,1>>,
NULL],
"-102m"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,-1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,1>>,
NULL],
"6/mmm"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,-1>>,
<<-1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,-1>>,
<<1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,-1>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,-1>>,
<<1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,-1>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,-1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
NULL],
"8/mmm"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
<<1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<1/2*2^(1/2),1/2*2^(1/2),0>|<1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,-1>>,
<<0,1,0>|<1,0,0>|<0,0,-1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,-1>>,
<<0,-1,0>|<-1,0,0>|<0,0,-1>>,
<<1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,-1>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,-1>>,
<<0,-1,0>|<1,0,0>|<0,0,-1>>,
<<1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,-1>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,-1>>,
<<0,1,0>|<-1,0,0>|<0,0,-1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
<<0,-1,0>|<-1,0,0>|<0,0,1>>,
<<1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<1/2*2^(1/2),1/2*2^(1/2),0>|<1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<0,1,0>|<1,0,0>|<0,0,1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
NULL],
"mm2"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
NULL],
"3m"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
NULL],
"4mm"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<0,-1,0>|<-1,0,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,1,0>|<1,0,0>|<0,0,1>>,
NULL],
"5m"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),-1/4*5^(1/2)-1/4,0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,1/4*(10-2*5^(1/2))^(1/2),0>|<1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,-1/4*(10-2*5^(1/2))^(1/2),0>|<-1/4*(10-2*5^(1/2))^(1/2),1/4+1/4*5^(1/2),0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),-1/4*(10+2*5^(1/2))^(1/2),0>|<-1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/4+1/4*5^(1/2),1/4*(10+2*5^(1/2))^(1/2),0>|<1/4*(10+2*5^(1/2))^(1/2),-1/4*5^(1/2)+1/4,0>|<0,0,1>>,
NULL],
"6mm"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<1/2,-1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),1/2,0>|<0,0,1>>,
<<1/2,-1/2*3^(1/2),0>|<-1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),-1/2,0>|<0,0,1>>,
<<-1/2,1/2*3^(1/2),0>|<1/2*3^(1/2),1/2,0>|<0,0,1>>,
NULL],
"8mm"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
<<1/2*2^(1/2),-1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
<<0,-1,0>|<-1,0,0>|<0,0,1>>,
<<1/2*2^(1/2),-1/2*2^(1/2),0>|<-1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<1/2*2^(1/2),1/2*2^(1/2),0>|<1/2*2^(1/2),-1/2*2^(1/2),0>|<0,0,1>>,
<<0,1,0>|<1,0,0>|<0,0,1>>,
<<-1/2*2^(1/2),1/2*2^(1/2),0>|<1/2*2^(1/2),1/2*2^(1/2),0>|<0,0,1>>,
NULL],
"23"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,1,0>|<0,0,1>|<1,0,0>>,
<<0,0,1>|<1,0,0>|<0,1,0>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,-1,0>|<0,0,-1>|<1,0,0>>,
<<0,0,-1>|<-1,0,0>|<0,1,0>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<0,1,0>|<0,0,-1>|<-1,0,0>>,
<<0,0,1>|<-1,0,0>|<0,-1,0>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,-1,0>|<0,0,1>|<-1,0,0>>,
<<0,0,-1>|<1,0,0>|<0,-1,0>>,
NULL],
"m-3"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,-1,0>|<0,0,-1>|<-1,0,0>>,
<<0,0,1>|<1,0,0>|<0,1,0>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<0,1,0>|<0,0,1>|<1,0,0>>,
<<0,0,-1>|<-1,0,0>|<0,-1,0>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,1,0>|<0,0,1>|<-1,0,0>>,
<<0,0,-1>|<-1,0,0>|<0,1,0>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,-1,0>|<0,0,-1>|<1,0,0>>,
<<0,0,1>|<1,0,0>|<0,-1,0>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<0,1,0>|<0,0,-1>|<-1,0,0>>,
<<0,0,-1>|<1,0,0>|<0,1,0>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<0,-1,0>|<0,0,1>|<1,0,0>>,
<<0,0,1>|<-1,0,0>|<0,-1,0>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,-1,0>|<0,0,1>|<-1,0,0>>,
<<0,0,1>|<-1,0,0>|<0,1,0>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,1,0>|<0,0,-1>|<1,0,0>>,
<<0,0,-1>|<1,0,0>|<0,-1,0>>,
NULL],
"-43m"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,-1>>,
<<0,0,-1>|<0,1,0>|<-1,0,0>>,
<<0,0,1>|<1,0,0>|<0,1,0>>,
<<0,0,-1>|<0,-1,0>|<1,0,0>>,
<<0,0,1>|<-1,0,0>|<0,-1,0>>,
<<0,0,-1>|<-1,0,0>|<0,1,0>>,
<<0,0,1>|<0,1,0>|<1,0,0>>,
<<0,0,-1>|<1,0,0>|<0,-1,0>>,
<<0,0,1>|<0,-1,0>|<-1,0,0>>,
<<1,0,0>|<0,0,1>|<0,1,0>>,
<<0,-1,0>|<0,0,-1>|<1,0,0>>,
<<-1,0,0>|<0,0,1>|<0,-1,0>>,
<<0,1,0>|<0,0,-1>|<-1,0,0>>,
<<0,-1,0>|<0,0,1>|<-1,0,0>>,
<<-1,0,0>|<0,0,-1>|<0,1,0>>,
<<0,1,0>|<0,0,1>|<1,0,0>>,
<<1,0,0>|<0,0,-1>|<0,-1,0>>,
<<0,-1,0>|<-1,0,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,1,0>|<1,0,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
NULL],
"432"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
<<0,0,1>|<0,-1,0>|<1,0,0>>,
<<0,0,1>|<1,0,0>|<0,1,0>>,
<<0,0,1>|<0,1,0>|<-1,0,0>>,
<<0,0,1>|<-1,0,0>|<0,-1,0>>,
<<0,0,-1>|<-1,0,0>|<0,1,0>>,
<<0,0,-1>|<0,-1,0>|<-1,0,0>>,
<<0,0,-1>|<1,0,0>|<0,-1,0>>,
<<0,0,-1>|<0,1,0>|<1,0,0>>,
<<-1,0,0>|<0,0,-1>|<0,-1,0>>,
<<0,-1,0>|<0,0,-1>|<1,0,0>>,
<<1,0,0>|<0,0,-1>|<0,1,0>>,
<<0,1,0>|<0,0,-1>|<-1,0,0>>,
<<0,-1,0>|<0,0,1>|<-1,0,0>>,
<<1,0,0>|<0,0,1>|<0,-1,0>>,
<<0,1,0>|<0,0,1>|<1,0,0>>,
<<-1,0,0>|<0,0,1>|<0,1,0>>,
<<0,1,0>|<1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,-1,0>|<-1,0,0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
NULL],
"m-3m"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<0,-1,0>|<1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,1,0>|<-1,0,0>|<0,0,-1>>,
<<0,0,1>|<0,-1,0>|<1,0,0>>,
<<0,0,-1>|<-1,0,0>|<0,-1,0>>,
<<0,0,1>|<0,1,0>|<-1,0,0>>,
<<0,0,-1>|<1,0,0>|<0,1,0>>,
<<0,0,1>|<1,0,0>|<0,-1,0>>,
<<0,0,-1>|<0,-1,0>|<-1,0,0>>,
<<0,0,1>|<-1,0,0>|<0,1,0>>,
<<0,0,-1>|<0,1,0>|<1,0,0>>,
<<1,0,0>|<0,0,1>|<0,1,0>>,
<<0,-1,0>|<0,0,-1>|<1,0,0>>,
<<-1,0,0>|<0,0,1>|<0,-1,0>>,
<<0,1,0>|<0,0,-1>|<-1,0,0>>,
<<0,-1,0>|<0,0,1>|<-1,0,0>>,
<<-1,0,0>|<0,0,-1>|<0,1,0>>,
<<0,1,0>|<0,0,1>|<1,0,0>>,
<<1,0,0>|<0,0,-1>|<0,-1,0>>,
<<0,1,0>|<1,0,0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<0,-1,0>|<-1,0,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<0,1,0>|<-1,0,0>|<0,0,1>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,-1,0>|<1,0,0>|<0,0,1>>,
<<0,0,-1>|<0,1,0>|<-1,0,0>>,
<<0,0,1>|<1,0,0>|<0,1,0>>,
<<0,0,-1>|<0,-1,0>|<1,0,0>>,
<<0,0,1>|<-1,0,0>|<0,-1,0>>,
<<0,0,-1>|<-1,0,0>|<0,1,0>>,
<<0,0,1>|<0,1,0>|<1,0,0>>,
<<0,0,-1>|<1,0,0>|<0,-1,0>>,
<<0,0,1>|<0,-1,0>|<-1,0,0>>,
<<-1,0,0>|<0,0,-1>|<0,-1,0>>,
<<0,1,0>|<0,0,1>|<-1,0,0>>,
<<1,0,0>|<0,0,-1>|<0,1,0>>,
<<0,-1,0>|<0,0,1>|<1,0,0>>,
<<0,1,0>|<0,0,-1>|<1,0,0>>,
<<1,0,0>|<0,0,1>|<0,-1,0>>,
<<0,-1,0>|<0,0,-1>|<-1,0,0>>,
<<-1,0,0>|<0,0,1>|<0,1,0>>,
<<0,-1,0>|<-1,0,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<0,1,0>|<1,0,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
NULL],
"25"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>>,
<<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>>,
<<0,1,0>|<0,0,1>|<1,0,0>>,
<<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>>,
<<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>>,
<<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>>,
<<0,0,1>|<1,0,0>|<0,1,0>>,
<<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>>,
<<0,0,-1>|<1,0,0>|<0,-1,0>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>>,
<<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>>,
<<0,1,0>|<0,0,-1>|<-1,0,0>>,
<<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>>,
<<0,-1,0>|<0,0,-1>|<1,0,0>>,
<<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>>,
<<0,0,-1>|<-1,0,0>|<0,1,0>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>>,
<<0,0,1>|<-1,0,0>|<0,-1,0>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>>,
<<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>>,
<<0,-1,0>|<0,0,1>|<-1,0,0>>,
<<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
NULL],
"m-5"=[
<<1,0,0>|<0,1,0>|<0,0,1>>,
<<1,0,0>|<0,-1,0>|<0,0,-1>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>>,
<<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>>,
<<0,1,0>|<0,0,1>|<1,0,0>>,
<<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>>,
<<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>>,
<<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>>,
<<0,0,1>|<1,0,0>|<0,1,0>>,
<<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>>,
<<0,0,-1>|<1,0,0>|<0,-1,0>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>>,
<<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>>,
<<0,1,0>|<0,0,-1>|<-1,0,0>>,
<<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<-1,0,0>|<0,-1,0>|<0,0,1>>,
<<-1,0,0>|<0,1,0>|<0,0,-1>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>>,
<<0,-1,0>|<0,0,-1>|<1,0,0>>,
<<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>>,
<<0,0,-1>|<-1,0,0>|<0,1,0>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>>,
<<0,0,1>|<-1,0,0>|<0,-1,0>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>>,
<<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>>,
<<0,-1,0>|<0,0,1>|<-1,0,0>>,
<<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<-1,0,0>|<0,-1,0>|<0,0,-1>>,
<<-1,0,0>|<0,1,0>|<0,0,1>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>>,
<<0,-1,0>|<0,0,-1>|<-1,0,0>>,
<<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>>,
<<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>>,
<<0,0,-1>|<-1,0,0>|<0,-1,0>>,
<<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>>,
<<0,0,1>|<-1,0,0>|<0,1,0>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>>,
<<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>>,
<<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>>,
<<0,-1,0>|<0,0,1>|<1,0,0>>,
<<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>>,
<<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<1,0,0>|<0,1,0>|<0,0,-1>>,
<<1,0,0>|<0,-1,0>|<0,0,1>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>>,
<<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<-1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,-1/2>>,
<<0,1,0>|<0,0,1>|<-1,0,0>>,
<<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4,1/2>>,
<<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4*5^(1/2)+1/4,-1/2,-1/4*5^(1/2)-1/4>>,
<<-1/4+1/4*5^(1/2),-1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),-1/2>>,
<<-1/4*5^(1/2)+1/4,1/2,-1/4*5^(1/2)-1/4>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<-1/4*5^(1/2)-1/4,-1/4+1/4*5^(1/2),1/2>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4*5^(1/2)+1/4,-1/2,1/4+1/4*5^(1/2)>>,
<<0,0,1>|<1,0,0>|<0,-1,0>>,
<<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>>,
<<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>>,
<<0,0,-1>|<1,0,0>|<0,1,0>>,
<<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>|<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>>,
<<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>>,
<<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>|<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>>,
<<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>|<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,-1/2>|<-1/4+1/4*5^(1/2),-1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2)>>,
<<-1/4+1/4*5^(1/2),1/2,-1/4*5^(1/2)-1/4>|<1/2,-1/4*5^(1/2)-1/4,-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),1/2>>,
<<0,1,0>|<0,0,-1>|<1,0,0>>,
<<-1/4+1/4*5^(1/2),1/2,1/4+1/4*5^(1/2)>|<-1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>|<1/4+1/4*5^(1/2),-1/4+1/4*5^(1/2),-1/2>>,
<<1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4,1/2>|<-1/4*5^(1/2)+1/4,1/2,1/4+1/4*5^(1/2)>|<1/2,1/4+1/4*5^(1/2),-1/4*5^(1/2)+1/4>>,
NULL],

# Space groups from http://www.cryst.ehu.es
# Note that old names Abm2,Aba2,Cmca,Cmma,Ccca are used for groups Aem2,Aea2,Cmce,Cmme,Ccce
"P1"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
NULL],
"P-1"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
NULL],
"P2"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
NULL],
"P21"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,0>],
NULL],
"C2"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"Pm"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
NULL],
"Pc"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
NULL],
"Cm"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
NULL],
"Cc"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
NULL],
"P2/m"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
NULL],
"P21/m"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,0>],
NULL],
"C2/m"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
NULL],
"P2/c"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
NULL],
"P21/c"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
NULL],
"C2/c"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
NULL],
"P222"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
NULL],
"P2221"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
NULL],
"P21212"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"P212121"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"C2221"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"C222"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"F222"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"I222"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
NULL],
"I212121"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,1/2>],
NULL],
"Pmm2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
NULL],
"Pmc21"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
NULL],
"Pcc2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,1/2>],
NULL],
"Pma2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,0>],
NULL],
"Pca21"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,1/2>],
NULL],
"Pnc2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
NULL],
"Pmn21"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
NULL],
"Pba2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
NULL],
"Pna21"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
NULL],
"Pnn2"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
NULL],
"Cmm2"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
NULL],
"Cmc21"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
NULL],
"Ccc2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
NULL],
"Amm2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
NULL],
"Abm2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,1/2>],
NULL],
"Ama2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
NULL],
"Aba2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,1/2>],
NULL],
"Fmm2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
NULL],
"Fdd2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/4,1/4,1/4>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/4,1/4,1/4>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/4,3/4,3/4>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/4,3/4,3/4>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<3/4,1/4,3/4>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<3/4,1/4,3/4>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<3/4,3/4,1/4>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<3/4,3/4,1/4>],
NULL],
"Imm2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
NULL],
"Iba2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,0>],
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[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,1/2>],
NULL],
"Ima2"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
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[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
NULL],
"Pmmm"=[
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[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
NULL],
"Pnnn"=[
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[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
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[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,1/2>],
NULL],
"Pccm"=[
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[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,1/2>],
NULL],
"Pban"=[
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[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,0>],
NULL],
"Pmma"=[
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[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<0,-1,0>|<0,0,-1>|<-1,0,0>>,<0,0,0>],
[<<0,1,0>|<0,0,1>|<-1,0,0>>,<0,0,0>],
[<<0,1,0>|<0,0,-1>|<1,0,0>>,<0,0,0>],
[<<0,-1,0>|<0,0,1>|<1,0,0>>,<0,0,0>],
[<<0,0,-1>|<-1,0,0>|<0,-1,0>>,<0,0,0>],
[<<0,0,1>|<1,0,0>|<0,-1,0>>,<0,0,0>],
[<<0,0,1>|<-1,0,0>|<0,1,0>>,<0,0,0>],
[<<0,0,-1>|<1,0,0>|<0,1,0>>,<0,0,0>],
[<<0,-1,0>|<-1,0,0>|<0,0,1>>,<0,0,0>],
[<<0,1,0>|<1,0,0>|<0,0,1>>,<0,0,0>],
[<<0,1,0>|<-1,0,0>|<0,0,-1>>,<0,0,0>],
[<<0,-1,0>|<1,0,0>|<0,0,-1>>,<0,0,0>],
[<<-1,0,0>|<0,0,1>|<0,-1,0>>,<0,0,0>],
[<<1,0,0>|<0,0,-1>|<0,-1,0>>,<0,0,0>],
[<<1,0,0>|<0,0,1>|<0,1,0>>,<0,0,0>],
[<<-1,0,0>|<0,0,-1>|<0,1,0>>,<0,0,0>],
[<<0,0,1>|<0,-1,0>|<-1,0,0>>,<0,0,0>],
[<<0,0,-1>|<0,1,0>|<-1,0,0>>,<0,0,0>],
[<<0,0,-1>|<0,-1,0>|<1,0,0>>,<0,0,0>],
[<<0,0,1>|<0,1,0>|<1,0,0>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<0,1,0>|<0,0,1>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<0,0,-1>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<0,0,1>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,1,0>|<0,0,-1>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<1,0,0>|<0,1,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<-1,0,0>|<0,1,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<1,0,0>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<-1,0,0>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<0,1,0>|<1,0,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<-1,0,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<1,0,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<0,1,0>|<-1,0,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,0,-1>|<0,1,0>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,0,1>|<0,1,0>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,0,-1>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,0,1>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<0,1,0>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<0,-1,0>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<0,1,0>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<0,-1,0>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<0,0,-1>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,1,0>|<0,0,1>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,1,0>|<0,0,-1>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<0,0,1>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<-1,0,0>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<1,0,0>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<-1,0,0>|<0,1,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<1,0,0>|<0,1,0>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<-1,0,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<0,1,0>|<1,0,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<0,1,0>|<-1,0,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<1,0,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,0,1>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,0,-1>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,0,1>|<0,1,0>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,0,-1>|<0,1,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<0,-1,0>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<0,1,0>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<0,-1,0>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<0,1,0>|<1,0,0>>,<1/2,1/2,1/2>],
NULL],
"Ia-3d"=[
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
[<<0,1,0>|<0,0,1>|<1,0,0>>,<0,0,0>],
[<<0,-1,0>|<0,0,-1>|<1,0,0>>,<1/2,1/2,0>],
[<<0,-1,0>|<0,0,1>|<-1,0,0>>,<1/2,0,1/2>],
[<<0,1,0>|<0,0,-1>|<-1,0,0>>,<0,1/2,1/2>],
[<<0,0,1>|<1,0,0>|<0,1,0>>,<0,0,0>],
[<<0,0,-1>|<-1,0,0>|<0,1,0>>,<0,1/2,1/2>],
[<<0,0,-1>|<1,0,0>|<0,-1,0>>,<1/2,1/2,0>],
[<<0,0,1>|<-1,0,0>|<0,-1,0>>,<1/2,0,1/2>],
[<<0,1,0>|<1,0,0>|<0,0,-1>>,<3/4,1/4,1/4>],
[<<0,-1,0>|<-1,0,0>|<0,0,-1>>,<3/4,3/4,3/4>],
[<<0,-1,0>|<1,0,0>|<0,0,1>>,<1/4,1/4,3/4>],
[<<0,1,0>|<-1,0,0>|<0,0,1>>,<1/4,3/4,1/4>],
[<<1,0,0>|<0,0,-1>|<0,1,0>>,<3/4,1/4,1/4>],
[<<-1,0,0>|<0,0,1>|<0,1,0>>,<1/4,3/4,1/4>],
[<<-1,0,0>|<0,0,-1>|<0,-1,0>>,<3/4,3/4,3/4>],
[<<1,0,0>|<0,0,1>|<0,-1,0>>,<1/4,1/4,3/4>],
[<<0,0,-1>|<0,1,0>|<1,0,0>>,<3/4,1/4,1/4>],
[<<0,0,1>|<0,-1,0>|<1,0,0>>,<1/4,1/4,3/4>],
[<<0,0,1>|<0,1,0>|<-1,0,0>>,<1/4,3/4,1/4>],
[<<0,0,-1>|<0,-1,0>|<-1,0,0>>,<3/4,3/4,3/4>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<0,-1,0>|<0,0,-1>|<-1,0,0>>,<0,0,0>],
[<<0,1,0>|<0,0,1>|<-1,0,0>>,<1/2,1/2,0>],
[<<0,1,0>|<0,0,-1>|<1,0,0>>,<1/2,0,1/2>],
[<<0,-1,0>|<0,0,1>|<1,0,0>>,<0,1/2,1/2>],
[<<0,0,-1>|<-1,0,0>|<0,-1,0>>,<0,0,0>],
[<<0,0,1>|<1,0,0>|<0,-1,0>>,<0,1/2,1/2>],
[<<0,0,1>|<-1,0,0>|<0,1,0>>,<1/2,1/2,0>],
[<<0,0,-1>|<1,0,0>|<0,1,0>>,<1/2,0,1/2>],
[<<0,-1,0>|<-1,0,0>|<0,0,1>>,<1/4,3/4,3/4>],
[<<0,1,0>|<1,0,0>|<0,0,1>>,<1/4,1/4,1/4>],
[<<0,1,0>|<-1,0,0>|<0,0,-1>>,<3/4,3/4,1/4>],
[<<0,-1,0>|<1,0,0>|<0,0,-1>>,<3/4,1/4,3/4>],
[<<-1,0,0>|<0,0,1>|<0,-1,0>>,<1/4,3/4,3/4>],
[<<1,0,0>|<0,0,-1>|<0,-1,0>>,<3/4,1/4,3/4>],
[<<1,0,0>|<0,0,1>|<0,1,0>>,<1/4,1/4,1/4>],
[<<-1,0,0>|<0,0,-1>|<0,1,0>>,<3/4,3/4,1/4>],
[<<0,0,1>|<0,-1,0>|<-1,0,0>>,<1/4,3/4,3/4>],
[<<0,0,-1>|<0,1,0>|<-1,0,0>>,<3/4,3/4,1/4>],
[<<0,0,-1>|<0,-1,0>|<1,0,0>>,<3/4,1/4,3/4>],
[<<0,0,1>|<0,1,0>|<1,0,0>>,<1/4,1/4,1/4>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,1>>,<0,1/2,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,1/2>],
[<<0,1,0>|<0,0,1>|<1,0,0>>,<1/2,1/2,1/2>],
[<<0,-1,0>|<0,0,-1>|<1,0,0>>,<0,0,1/2>],
[<<0,-1,0>|<0,0,1>|<-1,0,0>>,<0,1/2,0>],
[<<0,1,0>|<0,0,-1>|<-1,0,0>>,<1/2,0,0>],
[<<0,0,1>|<1,0,0>|<0,1,0>>,<1/2,1/2,1/2>],
[<<0,0,-1>|<-1,0,0>|<0,1,0>>,<1/2,0,0>],
[<<0,0,-1>|<1,0,0>|<0,-1,0>>,<0,0,1/2>],
[<<0,0,1>|<-1,0,0>|<0,-1,0>>,<0,1/2,0>],
[<<0,1,0>|<1,0,0>|<0,0,-1>>,<1/4,3/4,3/4>],
[<<0,-1,0>|<-1,0,0>|<0,0,-1>>,<1/4,1/4,1/4>],
[<<0,-1,0>|<1,0,0>|<0,0,1>>,<3/4,3/4,1/4>],
[<<0,1,0>|<-1,0,0>|<0,0,1>>,<3/4,1/4,3/4>],
[<<1,0,0>|<0,0,-1>|<0,1,0>>,<1/4,3/4,3/4>],
[<<-1,0,0>|<0,0,1>|<0,1,0>>,<3/4,1/4,3/4>],
[<<-1,0,0>|<0,0,-1>|<0,-1,0>>,<1/4,1/4,1/4>],
[<<1,0,0>|<0,0,1>|<0,-1,0>>,<3/4,3/4,1/4>],
[<<0,0,-1>|<0,1,0>|<1,0,0>>,<1/4,3/4,3/4>],
[<<0,0,1>|<0,-1,0>|<1,0,0>>,<3/4,3/4,1/4>],
[<<0,0,1>|<0,1,0>|<-1,0,0>>,<3/4,1/4,3/4>],
[<<0,0,-1>|<0,-1,0>|<-1,0,0>>,<1/4,1/4,1/4>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,1/2>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,1>>,<0,0,1/2>],
[<<0,-1,0>|<0,0,-1>|<-1,0,0>>,<1/2,1/2,1/2>],
[<<0,1,0>|<0,0,1>|<-1,0,0>>,<0,0,1/2>],
[<<0,1,0>|<0,0,-1>|<1,0,0>>,<0,1/2,0>],
[<<0,-1,0>|<0,0,1>|<1,0,0>>,<1/2,0,0>],
[<<0,0,-1>|<-1,0,0>|<0,-1,0>>,<1/2,1/2,1/2>],
[<<0,0,1>|<1,0,0>|<0,-1,0>>,<1/2,0,0>],
[<<0,0,1>|<-1,0,0>|<0,1,0>>,<0,0,1/2>],
[<<0,0,-1>|<1,0,0>|<0,1,0>>,<0,1/2,0>],
[<<0,-1,0>|<-1,0,0>|<0,0,1>>,<3/4,1/4,1/4>],
[<<0,1,0>|<1,0,0>|<0,0,1>>,<3/4,3/4,3/4>],
[<<0,1,0>|<-1,0,0>|<0,0,-1>>,<1/4,1/4,3/4>],
[<<0,-1,0>|<1,0,0>|<0,0,-1>>,<1/4,3/4,1/4>],
[<<-1,0,0>|<0,0,1>|<0,-1,0>>,<3/4,1/4,1/4>],
[<<1,0,0>|<0,0,-1>|<0,-1,0>>,<1/4,3/4,1/4>],
[<<1,0,0>|<0,0,1>|<0,1,0>>,<3/4,3/4,3/4>],
[<<-1,0,0>|<0,0,-1>|<0,1,0>>,<1/4,1/4,3/4>],
[<<0,0,1>|<0,-1,0>|<-1,0,0>>,<3/4,1/4,1/4>],
[<<0,0,-1>|<0,1,0>|<-1,0,0>>,<1/4,1/4,3/4>],
[<<0,0,-1>|<0,-1,0>|<1,0,0>>,<1/4,3/4,1/4>],
[<<0,0,1>|<0,1,0>|<1,0,0>>,<3/4,3/4,3/4>],
NULL],

# Layer groups from http://www.cryst.ehu.es (the setting for each layer group is set by the corresponding space group above)
"p1"="P1",
"p-1"="P-1",
"p112"="P112",
"p11m"="P11m",
"p11a"="P11a",
"p112/m"="P112/m",
"p112/a"="P112/a",
"p211"="P211",
"p2111"="P2111",
"c211"="C211",
"pm11"="Pm11",
"pb11"="Pb11",
"cm11"="Cm11",
"p2/m11"="P2/m11",
"p21/m11"="P21/m11",
"p2/b11"="P2/b11",
"p21/b11"="P21/b11",
"c2/m11"="C2/m11",
"p222"="P222",
"p2122"="P2122",
"p21212"="P21212",
"c222"="C222",
"pmm2"="Pmm2",
"pma2"="Pma2",
"pba2"="Pba2",
"cmm2"="Cmm2",
"pm2m"="Pm2m",
"pm21b"="Pm21b",
"pb21m"="Pb21m",
"pb2b"="Pb2b",
"pm2a"="Pm2a",
"pm21n"="Pm21n",
"pb21a"="Pb21a",
"pb2n"="Pb2n",
"cm2m"="Cm2m",
"cm2e"="Cm2e",
"pmmm"="Pmmm",
"pmaa"="Pmaa",
"pban"="Pban",
"pmam"="Pmam",
"pmma"="Pmma",
"pman"="Pman",
"pbaa"="Pbaa",
"pbam"="Pbam",
"pbma"="Pbma",
"pmmn"="Pmmn",
"cmmm"="Cmmm",
"cmme"="Cmme",
"p4"="P4",
"p-4"="P-4",
"p4/m"="P4/m",
"p4/n"="P4/n",
"p422"="P422",
"p4212"="P4212",
"p4mm"="P4mm",
"p4bm"="P4bm",
"p-42m"="P-42m",
"p-421m"="P-421m",
"p-4m2"="P-4m2",
"p-4b2"="P-4b2",
"p4/mmm"="P4/mmm",
"p4/nbm"="P4/nbm",
"p4/mbm"="P4/mbm",
"p4/nmm"="P4/nmm",
"p3"="P3",
"p-3"="P-3",
"p312"="P312",
"p321"="P321",
"p3m1"="P3m1",
"p31m"="P31m",
"p-31m"="P-31m",
"p-3m1"="P-3m1",
"p6"="P6",
"p-6"="P-6",
"p6/m"="P6/m",
"p622"="P622",
"p6mm"="P6mm",
"p-6m2"="P-6m2",
"p-62m"="P-62m",
"p6/mmm"="P6/mmm",

# Rod groups from http://www.cryst.ehu.es (the setting for each rod group is set by the corresponding space group above)
"q1"="P1",
"q-1"="P-1",
"q211"="P211",
"qm11"="Pm11",
"qc11"="Pc11",
"q2/m11"="P2/m11",
"q2/c11"="P2/c11",
"q112"="P112",
"q1121"="P1121",
"q11m"="P11m",
"q112/m"="P112/m",
"q1121/m"="P1121/m",
"q222"="P222",
"q2221"="P2221",
"qmm2"="Pmm2",
"qcc2"="Pcc2",
"qmc21"="Pmc21",
"q2mm"="P2mm",
"q2cm"="P2cm",
"qmmm"="Pmmm",
"qccm"="Pccm",
"qmcm"="Pmcm",
"q4"="P4",
"q41"="P41",
"q42"="P42",
"q43"="P43",
"q-4"="P-4",
"q4/m"="P4/m",
"q42/m"="P42/m",
"q422"="P422",
"q4122"="P4122",
"q4222"="P4222",
"q4322"="P4322",
"q4mm"="P4mm",
"q42cm"="P42cm",
"q4cc"="P4cc",
"q-42m"="P-42m",
"q-42c"="P-42c",
"q4/mmm"="P4/mmm",
"q4/mcc"="P4/mcc",
"q42/mmc"="P42/mmc",
"q3"="P3",
"q31"="P31",
"q32"="P32",
"q-3"="P-3",
"q312"="P312",
"q3112"="P3112",
"q3212"="P3212",
"q3m1"="P3m1",
"q3c1"="P3c1",
"q-31m"="P-31m",
"q-31c"="P-31c",
"q6"="P6",
"q61"="P61",
"q62"="P62",
"q63"="P63",
"q64"="P64",
"q65"="P65",
"q-6"="P-6",
"q6/m"="P6/m",
"q63/m"="P63/m",
"q622"="P622",
"q6122"="P6122",
"q6222"="P6222",
"q6322"="P6322",
"q6422"="P6422",
"q6522"="P6522",
"q6mm"="P6mm",
"q6cc"="P6cc",
"q63mc"="P63mc",
"q-6m2"="P-6m2",
"q-6c2"="P-6c2",
"q6/mmm"="P6/mmm",
"q6/mcc"="P6/mcc",
"q6/mmc"="P63/mmc",

# Nonstandard groups (reducible to one of the above groups)
""=[], # empty group
"A2/m"=[ # P2/m in double cell
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<0,0,0>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,0,1/2>],
[<<-1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,0,1/2>],
[<<1,0,0>|<0,-1,0>|<0,0,1>>,<1/2,0,1/2>],
NULL],
"b11a"=[ # p11a in double cell
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,0>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"c11a"=[ # p11a in double cell
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<1/2,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,1,0>|<0,0,-1>>,<0,1/2,0>],
NULL],
"b2111"=[ # p2111 in double cell
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,1/2,0>],
NULL],
"c2111"=[ # p2111 in double cell
[<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<1/2,0,0>],
[<<1,0,0>|<0,1,0>|<0,0,1>>,<1/2,1/2,0>],
[<<1,0,0>|<0,-1,0>|<0,0,-1>>,<0,1/2,0>],
NULL],

NULL]:# end of tbG
lsG:=map(lhs,tbG):
tbG:=table(tbG):

i:=SearchPos(lsG,"P1"):  PointGroupName:=lsG[..i-1]:  lsG:=lsG[i..]:
i:=SearchPos(lsG,"p1"):  SpaceGroupName:=lsG[..i-1]:  lsG:=lsG[i..]:
i:=SearchPos(lsG,"q1"):  LayerGroupName:=lsG[..i-1]:  lsG:=lsG[i..]:
i:=SearchPos(lsG,""):      RodGroupName:=lsG[..i-1]:  lsG:=lsG[i..]:
                         OtherGroupName:=lsG[2..]:
#hfl: Tables
SpaceGroupNumber:=table([seq(SpaceGroupName[i]=i,i=1..nops(SpaceGroupName))]):
#hfl: Tables
LayerGroupNumber:=table([seq(LayerGroupName[i]=i,i=1..nops(LayerGroupName))]):
#hfl: Tables
RodGroupNumber:=table([seq(RodGroupName[i]=i,i=1..nops(RodGroupName))]):

v:=[1,2,3,5,6,8,10,12,16,21,22,23,25,35,38,42,44,47,65,69,71,75,79,81,82,83,87,89,97,99,107,111,115,119,121,123,139,143,146,147,148,149,150,155,156,157,160,162,164,166,168,174,175,177,183,187,189,191,195,196,197,200,202,204,207,209,211,215,216,217,221,225,229,231]:
ArithmeticClass:=[seq(seq(SpaceGroupName[i]=SpaceGroupName[v[j]],i=v[j]..v[j+1]-1),j=1..nops(v)-1)]:
v:=[1,2,3,4,6,8,10,11,13,14,18,19,22,23,26,27,35,37,47,49,50,51,53,55,57,59,61,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81]:
ArithmeticClass:=[op(ArithmeticClass),seq(seq(LayerGroupName[i]=LayerGroupName[v[j]],i=v[j]..v[j+1]-1),j=1..nops(v)-1)]:
v:=[1,76]:
ArithmeticClass:=[op(ArithmeticClass),seq(seq(RodGroupName[i]=RodGroupName[v[j]],i=v[j]..v[j+1]-1),j=1..nops(v)-1)]:
#hfl: Tables
ArithmeticClass:=table([op(ArithmeticClass),seq(v=v,v=PointGroupName)]):

v:=[1,2,3,6,10,16,25,47,75,81,83,89,99,111,123,143,147,149,156,162,168,174,175,177,183,187,191,195,200,207,215,221,231]:
f:=G->piecewise(G="312","32",G="3m1","3m",G="-31m","-3m",G="-6m2","-62m",G):
GeometricClass:=[seq(seq(SpaceGroupName[i]=f(SpaceGroupName[v[j]][2..]),i=v[j]..v[j+1]-1),j=1..nops(v)-1)]:
v:=[1,2,3,4,6,8,11,14,19,23,27,37,49,50,51,53,55,57,61,65,66,67,69,71,73,74,75,76,77,78,80,81]:
f:=G->piecewise(G[..2]="11",G[3..],length(G)>2 and G[-2..]="11",cat(G[..-3],"x"),G="m2m","mm2y",G="312","32",G="3m1","3m",G="-31m","-3m",G="-6m2","-62m",G):
GeometricClass:=[op(GeometricClass),seq(seq(LayerGroupName[i]=f(LayerGroupName[v[j]][2..]),i=v[j]..v[j+1]-1),j=1..nops(v)-1)]:
v:=[1,76]:
f:=G->G:
GeometricClass:=[op(GeometricClass),seq(seq(RodGroupName[i]=f(RodGroupName[v[j]][2..]),i=v[j]..v[j+1]-1),j=1..nops(v)-1)]:
#hfl: Tables
GeometricClass:=table([op(GeometricClass),seq(v=v,v=PointGroupName)]):

#hfl: SymmetryGroup
TransformationTable:=table([

# Centering
"P"=<<1,0,0>|<0,1,0>|<0,0,1>>,           "p"=<<1,0,0>|<0,1,0>|<0,0,1>>,
"A"=<<1,0,0>|<0,1/2,-1/2>|<0,1/2,1/2>>,  "a"=<<1,0,0>|<0,1/2,-1/2>|<0,1/2,1/2>>,
"B"=<<1/2,0,-1/2>|<0,1,0>|<1/2,0,1/2>>,  "b"=<<1/2,0,-1/2>|<0,1,0>|<1/2,0,1/2>>,
"C"=<<1/2,-1/2,0>|<1/2,1/2,0>|<0,0,1>>,  "c"=<<1/2,-1/2,0>|<1/2,1/2,0>|<0,0,1>>,
"I"=<<-1/2,1/2,1/2>|<1/2,-1/2,1/2>|<1/2,1/2,-1/2>>,
"F"=<<0,1/2,1/2>|<1/2,0,1/2>|<1/2,1/2,0>>,
"R"=<<<2/3,1/3,1/3>|<-1/3,1/3,1/3>|-1/3,-2/3,1/3>>,

# Rotation
"xy"=<<1,0,0>|<0,1,0>|<0,0,1>>,  "z"=<<1,0,0>|<0,1,0>|<0,0,1>>,  # a,b,c in Bilbao (P matrix)
"yx"=<<0,1,0>|<1,0,0>|<0,0,1>>, "zw"=<<0,1,0>|<1,0,0>|<0,0,-1>>, # b,a,-c
"yz"=<<0,0,1>|<1,0,0>|<0,1,0>>,  "x"=<<0,0,1>|<1,0,0>|<0,1,0>>,  # c,a,b
"zy"=<<0,0,1>|<0,1,0>|<1,0,0>>, "yw"=<<0,0,-1>|<0,1,0>|<1,0,0>>, # -c,b,a
"zx"=<<0,1,0>|<0,0,1>|<1,0,0>>,  "y"=<<0,1,0>|<0,0,1>|<1,0,0>>,  # b,c,a
"xz"=<<1,0,0>|<0,0,1>|<0,1,0>>, "xw"=<<1,0,0>|<0,0,-1>|<0,1,0>>, # a,-c,b
"wz"=<<1/sqrt(2),1/sqrt(2),0>|<-1/sqrt(2),1/sqrt(2),0>|<0,0,1>>, # rotation by pi/4 in xy-plane
"wx"=<<0,0,1>|<1/sqrt(2),1/sqrt(2),0>|<-1/sqrt(2),1/sqrt(2),0>>, # rotation by pi/4 in xy-plane and z-axis to x
"wy"=<<-1/sqrt(2),1/sqrt(2),0>|<0,0,1>|<1/sqrt(2),1/sqrt(2),0>>, # rotation by pi/4 in xy-plane and z-axis to y
"w"=<<sqrt(2/3),0,sqrt(1/3)>|<-sqrt(1/6),sqrt(1/2),sqrt(1/3)>|<-sqrt(1/6),-sqrt(1/2),sqrt(1/3)>>, # z-axis to <1,1,1>

# Rotations not recognized in the symmetry group name
"r0"=<< 1/sqrt(2),-1/sqrt(2),0>|< 1/sqrt(6), 1/sqrt(6),-sqrt(2/3)>|<1/sqrt(3),1/sqrt(3),1/sqrt(3)>>, # rotate rhombohedral Tvs from hexagonal to cubic settings aligning them with axes
"r1"=<<-1/sqrt(2), 1/sqrt(2),0>|<-1/sqrt(6),-1/sqrt(6), sqrt(2/3)>|<1/sqrt(3),1/sqrt(3),1/sqrt(3)>>, # rotate rhombohedral Tvs from hexagonal to cubic settings aligning them with diagonals

# Alternative settings for space groups (T correspond to Bilbao's transformation matrix)
#3
"P121"="P2",
"P112"="P2x",
"P211"="P2y",
#4
"P1211"="P21",
"P1121"="P21x",
"P2111"="P21y",
#5
"C121"="C2",
"C211"=["C2",<<0,1,0>|<1,0,0>|<-1,0,-1>>],
#6
"P1m1"="Pm",
"P11m"="Pmx",
"Pm11"="Pmy",
#7
"Pn"=["Pc",<<-1,0,-1>|<0,1,0>|<1,0,0>>],
"P1c1"="Pc",
"P1n1"="Pn",
"P1a1"="Pczy",
"P11a"="Pcx",
"P11n"="Pnx",
"P11b"="Pcxz",
"Pb11"="Pcy",
"Pn11"="Pny",
"Pc11"="Pcyx",
#8
"C1m1"="Cm",
"Cm11"=["Cm",<<0,1,0>|<1,0,0>|<-1,0,-1>>],
#10
"P12/m1"="P2/m",
"P112/m"="P2/mx",
"P2/m11"="P2/my",
#11
"P121/m1"="P21/m",
"P1121/m"="P21/mx",
"P21/m11"="P21/my",
#12
"C12/m1"="C2/m",
"C2/m11"=["C2/m",<<0,1,0>|<1,0,0>|<-1,0,-1>>],
#13
"P12/c1"="P2/c",
"P112/a"="P2/cx",
"P2/b11"="P2/cy",
"P2/c11"=["P2/c",<<0,1,0>|<-1,0,-1>|<0,0,1>>],
#14
"P121/c1"="P21/c",
"P1121/a"="P21/cx",
"P21/b11"="P21/cy",
"P121/n1"=["P21/c",<<-1,0,-1>|<0,1,0>|<1,0,0>>],  "P21/n"=["P21/c",<<-1,0,-1>|<0,1,0>|<1,0,0>>],
"P121/a1"=["P21/c",<<0,0,1>|<0,1,0>|<-1,0,-1>>],  "P21/a"=["P21/c",<<0,0,1>|<0,1,0>|<-1,0,-1>>],
"P1121/n"=["P21/c",<<1,0,0>|<-1,0,-1>|<0,1,0>>],
"P1121/b"=["P21/c",<<-1,0,-1>|<0,0,1>|<0,1,0>>],
"P21/n11"=["P21/c",<<0,1,0>|<1,0,0>|<-1,0,-1>>],
"P21/c11"=["P21/c",<<0,1,0>|<-1,0,-1>|<0,0,1>>],
#15
"C12/c1"="C2/c",
"A12/n1"=["C2/c",<<-1,0,-1>|<0,1,0>|<1,0,0>>],
"I12/a1"=["C2/c",<<0,0,1>|<0,1,0>|<-1,0,-1>>],
"A12/a1"=["C2/c",<<0,0,1>|<0,-1,0>|<1,0,0>>],
"C12/n1"=["C2/c",<<1,0,0>|<0,-1,0>|<-1,0,-1>>],
"I12/c1"=["C2/c",<<-1,0,-1>|<0,-1,0>|<0,0,1>>],
"A112/a"=["C2/c",<<0,0,1>|<1,0,0>|<0,1,0>>],
"B112/n"=["C2/c",<<1,0,0>|<-1,0,-1>|<0,1,0>>],
"I112/b"=["C2/c",<<-1,0,-1>|<0,0,1>|<0,1,0>>],
"B112/b"=["C2/c",<<1,0,0>|<0,0,1>|<0,-1,0>>],
"A112/n"=["C2/c",<<-1,0,-1>|<1,0,0>|<0,-1,0>>],
"I112/a"=["C2/c",<<0,0,1>|<-1,0,-1>|<0,-1,0>>],
"B2/b11"=["C2/c",<<0,1,0>|<0,0,1>|<1,0,0>>],
"C2/n11"=["C2/c",<<0,1,0>|<1,0,0>|<-1,0,-1>>],
"I2/c11"=["C2/c",<<0,1,0>|<-1,0,-1>|<0,0,1>>],
"C2/c11"=["C2/c",<<0,-1,0>|<1,0,0>|<0,0,1>>],
"B2/n11"=["C2/c",<<0,-1,0>|<-1,0,-1>|<1,0,0>>],
"I2/b11"=["C2/c",<<0,-1,0>|<0,0,1>|<-1,0,-1>>],
#17
"P2122" ="P2221x",  "p2122:1"="p2122xz",  # Bilbao setting
"P2212" ="P2221y",
"P12221"="P2221yx",   # non-ITA
"P12122"="P2221zy",   # non-ITA
"P12212"="P2221xz",   # non-ITA
#25
"P2mm"="Pmm2x",
"Pm2m"="Pmm2y",
#26
"Pmc21"="Pmc21",
"P21ma"="Pmc21x",  "p21ma"="pm21bzy",
"Pb21m"="Pmc21y",
"Pm21b"="Pmc21xw",
#27
"Pb2b"="Pcc2y",
#28
"Pm2a"="Pma2xw",
"P2cm"="Pma2yw",   "q2cm:4"=["q2cm",<<1,0,0>|<0,1,0>|<0,0,1>>,<0,0,1/4>],
#29
"Pbc21"="Pca21yx",
"P21ab"="Pca21x",
"P21ca"="Pca21zy",
"Pc21b"="Pca21y",
"Pb21a"="Pca21xz",
#30
"Pb2n"="Pnc2y",
#31
"P21mn"="Pmn21x",  "p21mn"=["pm21n",<<0,0,1>|<0,1,0>|<-1,0,0>>],
"Pm21n"="Pmn21xw",
#33
"Pbn21"="Pna21yx",
"P21nb"="Pna21x",
"P21cn"="Pna21zy",
"Pc21n"="Pna21y",
"Pn21a"="Pna21xz",
#38
"Cm2m"="Amm2y",
#39
"Aem2"="Abm2",  # non-ITA but commonly used
"Cm2e"="Abm2y",
#41
"Aea2"="Aba2",  # non-ITA but commonly used
#49
"Pmaa"="Pccmx",
#51
"Pmcm"="Pmmay",
"Pmam"="Pmmaxw",
#53
"Pnmb"="Pmnazw",
"Pbmn"="Pmnax",   "pbmn"="pmanzw",
"Pcnm"="Pmnayw",
"Pncm"="Pmnay",
"Pman"="Pmnaxw",
#54
"Pbaa"="Pccax",
#57
"Pcam"="Pbcmzw",
"Pmca"="Pbcmx",
"Pmab"="Pbcmyw",
"Pbma"="Pbcmy",
"Pcmb"="Pbcmxw",
#62
"Pmnb"="Pnmazw",
"Pbnm"="Pnmax",
"Pcmn"="Pnmayw",
"Pmcn"="Pnmay",
"Pnam"="Pnmaxw",
#64
"Cmce"="Cmca",  # non-ITA but commonly used
"Abma"="Cmcax",  "Aema"="Cmcax",
"Bbcm"="Cmcay",  "Bbem"="Cmcay",
#65
"Ammm"="Cmmmx",
"Bmmm"="Cmmmy",
#67
"Cmme"="Cmma",  # non-ITA but commonly used
"Abmm"="Cmmax",  "Aemm"="Cmmax",
"Bmcm"="Cmmay",  "Bmem"="Cmmay",
#68
"Ccce"="Ccca",  # non-ITA but commonly used
#74
"Immb"="Immazw",
"Ibmm"="Immax",
"Icmm"="Immayw",
"Imcm"="Immay",
"Imam"="Immaxw",
#90
"p4212:1"="p4212:200",  # Bilbao setting
#91
"q4122:1"="q4122:004",  # Bilbao setting
#95
"q4322:1"="q4322:004",  # Bilbao setting
#141
"I41/amd:2"="I41/amd",
"I41/amd:1"=["I41/amd",<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/4,-1/8>],
#142
"I41/acd:2"="I41/acd",
"I41/acd:1"=["I41/acd",<<1,0,0>|<0,1,0>|<0,0,1>>,<0,1/4,-1/8>],
#188
"q-6c2:1"="q-6c2:004",  # Bilbao setting
#227
"Fd-3m:2"="Fd-3m",
"Fd-3m:1"=["Fd-3m",<<1,0,0>|<0,1,0>|<0,0,1>>,<-1/8,-1/8,-1/8>],

# Schoenflies names for point groups
seq(HM2Scho(i)=i,i=PointGroupName),
NULL]):# end of TransformationTable


unassign('k');
tbK:=table([            #  Bravais  #
"Pmmm"=[
 "G"=<0,0,0>,
 "X"=<1/2,0,0>,
 "Y"=<0,1/2,0>,
 "Z"=<0,0,1/2>,
 "T"=<0,1/2,1/2>,
 "U"=<1/2,0,1/2>,
 "S"=<1/2,1/2,0>,
 "R"=<1/2,1/2,1/2>,
"SM"=<k,0,0>,
"DT"=<0,k,0>,
"LD"=<0,0,k>,
"path"=["GXSR","GXSRTZG","GXSRTYG"],
"id"=[],
NULL],
"Immm"=[
 "G"=<0,0,0>,
 "H"=<1/2,1/2,-1/2>,
 "X"=<-1/2,1/2,1/2>,
 "Y"=<1/2,-1/2,1/2>,
 "Z"=<1/2,1/2,-1/2>,
 "S"=<1/2,0,0>,
 "R"=<0,1/2,0>,
 "T"=<0,0,1/2>,
 "P"=<1/4,1/4,1/4>,
"DX"=<-k,k,k>,
"DY"=<k,-k,k>,
"DZ"=<k,k,-k>,
"path"=["PGHS","PGX,ZGYS"],
"id"=["X"="H","Y"="H","Z"="H","W"="P","SM"="DX","DT"="DY","LD"="DZ"],
NULL],
"Pm-3m"=[
 "G"=<0,0,0>,
 "X"=<1/2,0,0>,
 "M"=<1/2,1/2,0>,
 "R"=<1/2,1/2,1/2>,
"DT"=<k,0,0>,
"SM"=<k,k,0>,
"LD"=<k,k,k>,
 "Z"=<1/2,k,0>,     # X-M at k=0..1/2
 "S"=<1/2,k,k>,     # X-R at k=0..1/2
 "T"=<1/2,1/2,k>,   # M-R at k=0..1/2
NULL],
"Fm-3m"=[
 "G"=<0,0,0>,           # 0,0,0     #
 "X"=<0,1/2,1/2>,       # 1,0,0     #
 "L"=<1/2,1/2,1/2>,     #1/2,1/2,1/2# also <1/2,0,0>
 "W"=<1/4,1/2,3/4>,     # 1,1/2,0   #
"DT"=<0,k,k>,           # 2*k,0,0   #
"LD"=<k,k,k>,           # k,k,k     # also <k,0,0>->(-k,k,k)
"SM"=<k,k,2*k>,         # 2*k,2*k,0 # G-K-M at k=0..3/8..1/2, also <-k,k,0>->(2*k,-2*k,0)
 "K"=<3/8,3/8,3/4>,     # 3/4,3/4,0 # =U
 "M"=<1/2,1/2,1>,       # 1,1,0     #
 "S"=<2*k,1/2+k,1/2+k>, # 1,2*k,2*k # X-U at k=0..1/8
 "U"=<1/4,5/8,5/8>,     # 1,1/4,1/4 # =K
 "V"=<k,1/2,1/2+k>,     # 1,2*k,0   # X-W at k=0..1/4
 "Q"=<1/4+k,1/2,3/4-k>, #1-2*k,1/2,2*k# W-L at k=0..1/4
 "R"=<1,1,1>,           # 1,1,1     # ="G"
 "P"=<1/2,0,0>,         #-1/2,1/2,1/2# ="L"
"PD"=<k,0,0>,           # -k,k,k    # ="LD"
"path"=["LGXR"], # diamond holes L-G-X-U=K-G  [<0,0,1/2>,<0,0,0>,<0,1/2,1/2>,<1,1,1>]
"id"=["R"="G","P"="L","U"="K","PD"="LD"],
NULL],
"Im-3m"=[
 "G"=<0,0,0>,              # 0,0,0     #
 "H"=<-1/2,1/2,1/2>,       # 1,0,0     # =X
 "P"=<1/4,1/4,1/4>,        #1/2,1/2,1/2# =L
 "N"=<0,0,1/2>,            # 1/2,1/2,0 # =K
"DT"=<-k,k,k>,             # 2*k,0,0   # G-H at k=0..1/2
"LD"=<k,k,k>,              #2*k,2*k,2*k# G-P at k=0..1/4
"SM"=<0,0,k>,              # k,k,0     # G-N-M at k=0..1/2..1
 "D"=<k,k,1/2-k>,          #1/2,1/2,2*k# N-P at k=0..1/4
"Gp"=<k-1/2,1/2-k,1/2>,    # 1-k,k,0   # H-N at k=0..1/2
 "F"=<3*k-1/2,1/2-k,1/2-k>,#1-2*k,2*k,2*k# H-P at k=0..1/4
"path"=["GPHGN"],
"id"=[],
NULL],
"pmmm"=[
 "G"=<0,0,0>,
 "X"=<1/2,0,0>,
 "Y"=<0,1/2,0>,
 "S"=<1/2,1/2,0>,
 "H"=<1/2,-1/2,0>,
"SM"=<k,0>,
"DT"=<0,k>,
"path"=["SYGXS","SGH"],
"id"=[],
NULL],
"p-3"=[
 "G"=<0,0>,
 "K"=<1/3,1/3>,
 "M"=<1/2,1/2>,   # =X
 "X"=<1/2,0>,     # =M
"LD"=<k,k>,       # G-K-M at k=0..1/3..1/2
"DT"=<k,0>,       # =SM
 "T"=<1/2-k,2*k>, # X-K at k=0..1/6
"path"=["XGKM"],
"id"=["X"="M"],
NULL],
"I222"="Immm",
"I432"="Im-3m",
"p-3m1"="p-3",
"p6/mmm"="p-3",
NULL]):# end of tbK

ModuleLoad()

end module: