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

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

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

```

to this polytope