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# Polytope of Type {2,3,6}

Atlas Canonical Name : {2,3,6}*216
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 4
Schlafli Type : {2,3,6}
Number of vertices, edges, etc : 2, 9, 27, 18
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,3,6,2} of size 432
{2,3,6,3} of size 648
{2,3,6,4} of size 864
{2,3,6,6} of size 1296
{2,3,6,6} of size 1296
{2,3,6,8} of size 1728
{2,3,6,3} of size 1944
{2,3,6,9} of size 1944
Vertex Figure Of :
{2,2,3,6} of size 432
{3,2,3,6} of size 648
{4,2,3,6} of size 864
{5,2,3,6} of size 1080
{6,2,3,6} of size 1296
{7,2,3,6} of size 1512
{8,2,3,6} of size 1728
{9,2,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,3,6}*72
9-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,6,6}*432c
3-fold covers : {2,9,6}*648a, {2,9,6}*648b, {2,9,6}*648c, {2,9,6}*648d, {2,3,6}*648, {2,3,18}*648, {6,3,6}*648a
4-fold covers : {2,12,6}*864a, {4,6,6}*864a, {2,6,12}*864c, {2,3,6}*864, {2,3,12}*864, {4,3,6}*864
5-fold covers : {2,15,6}*1080
6-fold covers : {2,18,6}*1296a, {2,18,6}*1296c, {2,18,6}*1296d, {2,18,6}*1296e, {2,6,6}*1296d, {2,6,18}*1296h, {6,6,6}*1296d, {6,6,6}*1296e, {2,6,6}*1296e
7-fold covers : {2,21,6}*1512
8-fold covers : {4,12,6}*1728a, {2,24,6}*1728a, {8,6,6}*1728a, {2,12,12}*1728a, {2,6,24}*1728c, {4,6,12}*1728c, {2,3,12}*1728, {2,3,24}*1728, {8,3,6}*1728, {4,6,6}*1728a, {2,6,6}*1728a, {2,6,12}*1728a
9-fold covers : {2,9,18}*1944a, {2,9,6}*1944a, {2,3,18}*1944a, {2,9,6}*1944b, {2,9,18}*1944b, {2,9,6}*1944c, {2,9,18}*1944c, {2,9,18}*1944d, {2,9,18}*1944e, {2,27,6}*1944a, {2,9,6}*1944d, {2,9,18}*1944f, {2,9,18}*1944g, {2,9,18}*1944h, {2,9,18}*1944i, {2,9,6}*1944e, {2,9,18}*1944j, {2,27,6}*1944b, {2,27,6}*1944c, {2,3,6}*1944, {2,3,18}*1944b, {6,9,6}*1944a, {6,3,6}*1944a, {6,3,6}*1944b, {6,9,6}*1944c, {6,9,6}*1944d, {6,9,6}*1944g, {6,3,6}*1944d, {6,3,18}*1944
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (4,5)(6,9)(7,8);;
s2 := ( 3, 6)( 4,11)( 7,10);;
s3 := ( 6, 7)( 8, 9)(10,11);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(11)!(1,2);
s1 := Sym(11)!(4,5)(6,9)(7,8);
s2 := Sym(11)!( 3, 6)( 4,11)( 7,10);
s3 := Sym(11)!( 6, 7)( 8, 9)(10,11);
poly := sub<Sym(11)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;

to this polytope