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

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

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

```

to this polytope