# Polytope of Type {6,3,6}

Atlas Canonical Name : {6,3,6}*648a
Also Known As : 6T4(1,1)(3,0)if this polytope has another name.
Group : SmallGroup(648,555)
Rank : 4
Schlafli Type : {6,3,6}
Number of vertices, edges, etc : 6, 27, 27, 18
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,3,6,2} of size 1296
{6,3,6,3} of size 1944
Vertex Figure Of :
{2,6,3,6} of size 1296
{3,6,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,3,6}*216, {6,3,6}*216
9-fold quotients : {2,3,6}*72, {6,3,2}*72
27-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6,6}*1296e
3-fold covers : {6,9,6}*1944a, {6,3,6}*1944a, {6,3,6}*1944b, {6,9,6}*1944c, {6,9,6}*1944d, {6,9,6}*1944g, {6,3,6}*1944d, {6,3,18}*1944
Permutation Representation (GAP) :
```s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);;
s1 := ( 1, 4)( 2, 5)( 3, 6)(10,22)(11,23)(12,24)(13,19)(14,20)(15,21)(16,25)
(17,26)(18,27);;
s2 := ( 1,21)( 2,19)( 3,20)( 4,27)( 5,25)( 6,26)( 7,24)( 8,22)( 9,23)(13,16)
(14,17)(15,18);;
s3 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27);;
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*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

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

```
References :
1. Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (\ Cambridge University Press, 2002)

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