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# Polytope of Type {2,2,3,6}

Atlas Canonical Name : {2,2,3,6}*144
if this polytope has a name.
Group : SmallGroup(144,192)
Rank : 5
Schlafli Type : {2,2,3,6}
Number of vertices, edges, etc : 2, 2, 3, 9, 6
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,2,3,6,2} of size 288
{2,2,3,6,3} of size 432
{2,2,3,6,4} of size 576
{2,2,3,6,6} of size 864
{2,2,3,6,6} of size 864
{2,2,3,6,8} of size 1152
{2,2,3,6,9} of size 1296
{2,2,3,6,3} of size 1296
{2,2,3,6,10} of size 1440
{2,2,3,6,12} of size 1728
{2,2,3,6,12} of size 1728
{2,2,3,6,4} of size 1728
Vertex Figure Of :
{2,2,2,3,6} of size 288
{3,2,2,3,6} of size 432
{4,2,2,3,6} of size 576
{5,2,2,3,6} of size 720
{6,2,2,3,6} of size 864
{7,2,2,3,6} of size 1008
{8,2,2,3,6} of size 1152
{9,2,2,3,6} of size 1296
{10,2,2,3,6} of size 1440
{11,2,2,3,6} of size 1584
{12,2,2,3,6} of size 1728
{13,2,2,3,6} of size 1872
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,2,3,6}*288, {2,2,6,6}*288c
3-fold covers : {2,2,9,6}*432, {2,2,3,6}*432, {2,6,3,6}*432, {6,2,3,6}*432
4-fold covers : {8,2,3,6}*576, {2,2,12,6}*576b, {2,4,6,6}*576b, {4,2,6,6}*576c, {2,2,6,12}*576c, {2,2,3,6}*576, {2,2,3,12}*576, {2,4,3,6}*576
5-fold covers : {10,2,3,6}*720, {2,2,15,6}*720
6-fold covers : {4,2,9,6}*864, {4,2,3,6}*864, {2,2,18,6}*864b, {2,2,6,6}*864c, {12,2,3,6}*864, {4,6,3,6}*864, {2,2,6,6}*864d, {2,6,6,6}*864e, {2,6,6,6}*864f, {6,2,6,6}*864c
7-fold covers : {14,2,3,6}*1008, {2,2,21,6}*1008
8-fold covers : {16,2,3,6}*1152, {4,4,6,6}*1152b, {2,4,12,6}*1152b, {2,2,12,12}*1152c, {4,2,6,12}*1152a, {4,2,12,6}*1152c, {2,4,6,12}*1152a, {2,8,6,6}*1152b, {8,2,6,6}*1152c, {2,2,6,24}*1152a, {2,2,24,6}*1152c, {4,4,3,6}*1152b, {4,2,3,6}*1152, {4,2,3,12}*1152, {2,2,3,12}*1152, {2,2,3,24}*1152, {2,8,3,6}*1152, {2,2,6,6}*1152b, {2,2,6,12}*1152b, {2,4,6,6}*1152b
9-fold covers : {2,2,9,18}*1296, {2,2,9,6}*1296a, {2,2,27,6}*1296, {2,2,9,6}*1296b, {2,2,9,6}*1296c, {2,2,9,6}*1296d, {2,2,3,6}*1296, {2,2,3,18}*1296, {2,6,9,6}*1296, {6,2,9,6}*1296, {18,2,3,6}*1296, {6,6,3,6}*1296a, {2,6,3,6}*1296a, {2,6,3,6}*1296b, {6,2,3,6}*1296, {6,6,3,6}*1296b
10-fold covers : {20,2,3,6}*1440, {4,2,15,6}*1440, {2,2,6,30}*1440a, {2,10,6,6}*1440c, {10,2,6,6}*1440c, {2,2,30,6}*1440c
11-fold covers : {22,2,3,6}*1584, {2,2,33,6}*1584
12-fold covers : {8,2,9,6}*1728, {8,2,3,6}*1728, {2,2,36,6}*1728b, {2,2,12,6}*1728a, {2,4,18,6}*1728b, {4,2,18,6}*1728b, {2,4,6,6}*1728a, {4,2,6,6}*1728c, {2,2,18,12}*1728b, {2,2,6,12}*1728c, {24,2,3,6}*1728, {8,6,3,6}*1728, {2,2,9,6}*1728, {2,2,9,12}*1728, {2,4,9,6}*1728, {2,2,3,6}*1728, {2,2,3,12}*1728, {2,4,3,6}*1728, {2,6,12,6}*1728d, {2,6,12,6}*1728e, {2,12,6,6}*1728c, {6,2,12,6}*1728b, {12,2,6,6}*1728c, {4,6,6,6}*1728e, {6,4,6,6}*1728b, {4,2,6,6}*1728d, {2,2,6,12}*1728g, {2,2,12,6}*1728g, {4,6,6,6}*1728h, {2,4,6,6}*1728h, {2,6,6,12}*1728f, {2,6,6,12}*1728g, {2,12,6,6}*1728g, {6,2,6,12}*1728c, {6,4,3,6}*1728, {2,6,3,6}*1728a, {2,6,3,6}*1728b, {2,6,3,12}*1728, {2,12,3,6}*1728, {6,2,3,6}*1728, {6,2,3,12}*1728
13-fold covers : {26,2,3,6}*1872, {2,2,39,6}*1872
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 8, 9)(10,13)(11,12);;
s3 := ( 5,10)( 6, 8)( 7,12)( 9,11);;
s4 := ( 8, 9)(10,11)(12,13);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(13)!(1,2);
s1 := Sym(13)!(3,4);
s2 := Sym(13)!( 6, 7)( 8, 9)(10,13)(11,12);
s3 := Sym(13)!( 5,10)( 6, 8)( 7,12)( 9,11);
s4 := Sym(13)!( 8, 9)(10,11)(12,13);
poly := sub<Sym(13)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3 >;

```

to this polytope