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# Polytope of Type {6,4,6}

Atlas Canonical Name : {6,4,6}*288
Also Known As : {{6,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(288,958)
Rank : 4
Schlafli Type : {6,4,6}
Number of vertices, edges, etc : 6, 12, 12, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,4,6,2} of size 576
{6,4,6,3} of size 864
{6,4,6,4} of size 1152
{6,4,6,3} of size 1152
{6,4,6,4} of size 1152
{6,4,6,6} of size 1728
{6,4,6,6} of size 1728
{6,4,6,6} of size 1728
Vertex Figure Of :
{2,6,4,6} of size 576
{3,6,4,6} of size 864
{4,6,4,6} of size 1152
{3,6,4,6} of size 1152
{4,6,4,6} of size 1152
{6,6,4,6} of size 1728
{6,6,4,6} of size 1728
{6,6,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,2,6}*144
3-fold quotients : {2,4,6}*96a, {6,4,2}*96a
4-fold quotients : {3,2,6}*72, {6,2,3}*72
6-fold quotients : {2,2,6}*48, {6,2,2}*48
8-fold quotients : {3,2,3}*36
9-fold quotients : {2,4,2}*32
12-fold quotients : {2,2,3}*24, {3,2,2}*24
18-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,4,12}*576, {12,4,6}*576, {6,8,6}*576
3-fold covers : {6,4,18}*864, {18,4,6}*864, {6,12,6}*864a, {6,12,6}*864b, {6,12,6}*864f, {6,12,6}*864g
4-fold covers : {12,4,12}*1152, {6,8,12}*1152a, {12,8,6}*1152a, {6,4,24}*1152a, {24,4,6}*1152a, {6,8,12}*1152b, {12,8,6}*1152b, {6,4,24}*1152b, {24,4,6}*1152b, {6,4,12}*1152a, {12,4,6}*1152a, {6,16,6}*1152, {6,4,6}*1152a, {6,4,6}*1152b
5-fold covers : {6,20,6}*1440, {6,4,30}*1440, {30,4,6}*1440
6-fold covers : {12,4,18}*1728, {18,4,12}*1728, {6,4,36}*1728, {36,4,6}*1728, {6,12,12}*1728a, {12,12,6}*1728a, {6,8,18}*1728, {18,8,6}*1728, {6,24,6}*1728a, {6,24,6}*1728b, {6,12,12}*1728b, {6,12,12}*1728c, {12,12,6}*1728b, {12,12,6}*1728f, {6,24,6}*1728f, {6,24,6}*1728g, {6,12,12}*1728g, {12,12,6}*1728g
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36);;
s1 := ( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,31)(20,32)(21,33)(22,28)
(23,29)(24,30)(25,34)(26,35)(27,36);;
s2 := ( 1,19)( 2,21)( 3,20)( 4,22)( 5,24)( 6,23)( 7,25)( 8,27)( 9,26)(10,28)
(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35);;
s3 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)
(31,32)(34,35);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(36)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36);
s1 := Sym(36)!( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,31)(20,32)(21,33)
(22,28)(23,29)(24,30)(25,34)(26,35)(27,36);
s2 := Sym(36)!( 1,19)( 2,21)( 3,20)( 4,22)( 5,24)( 6,23)( 7,25)( 8,27)( 9,26)
(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35);
s3 := Sym(36)!( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)
(28,29)(31,32)(34,35);
poly := sub<Sym(36)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope