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# Polytope of Type {6,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*288b
if this polytope has a name.
Group : SmallGroup(288,958)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 6, 18, 12, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,4,2} of size 576
{6,6,4,4} of size 1152
{6,6,4,6} of size 1728
{6,6,4,3} of size 1728
Vertex Figure Of :
{2,6,6,4} of size 576
{3,6,6,4} of size 864
{4,6,6,4} of size 1152
{6,6,6,4} of size 1728
{6,6,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,2}*144b
3-fold quotients : {2,6,4}*96a
4-fold quotients : {6,3,2}*72
6-fold quotients : {2,6,2}*48
9-fold quotients : {2,2,4}*32
12-fold quotients : {2,3,2}*24
18-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12,4}*576b, {6,6,8}*576b, {12,6,4}*576c
3-fold covers : {6,18,4}*864b, {6,6,4}*864a, {6,6,12}*864d, {6,6,4}*864h, {6,6,12}*864g
4-fold covers : {12,12,4}*1152a, {6,12,8}*1152a, {6,24,4}*1152b, {6,12,8}*1152d, {6,24,4}*1152e, {6,12,4}*1152a, {12,6,8}*1152a, {24,6,4}*1152a, {6,6,16}*1152a, {6,6,4}*1152d, {6,6,4}*1152e, {12,6,4}*1152c
5-fold covers : {6,6,20}*1440b, {30,6,4}*1440a, {6,30,4}*1440c
6-fold covers : {6,36,4}*1728b, {6,12,4}*1728a, {6,18,8}*1728b, {6,6,8}*1728a, {12,18,4}*1728b, {12,6,4}*1728c, {6,6,24}*1728d, {12,6,12}*1728c, {6,12,12}*1728d, {6,12,12}*1728f, {6,6,8}*1728e, {6,6,24}*1728g, {12,6,12}*1728g, {6,12,4}*1728j, {12,6,4}*1728h
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, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)
(21,23)(26,27)(28,31)(29,33)(30,32)(35,36);;
s2 := ( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,29)(20,28)
(21,30)(22,35)(23,34)(24,36)(25,32)(26,31)(27,33);;
s3 := ( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)
(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)
(20,24)(21,23)(26,27)(28,31)(29,33)(30,32)(35,36);
s2 := Sym(36)!( 1, 2)( 4, 8)( 5, 7)( 6, 9)(10,11)(13,17)(14,16)(15,18)(19,29)
(20,28)(21,30)(22,35)(23,34)(24,36)(25,32)(26,31)(27,33);
s3 := Sym(36)!( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)
(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36);
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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope