# Polytope of Type {6,6,3}

Atlas Canonical Name : {6,6,3}*648e
if this polytope has a name.
Group : SmallGroup(648,555)
Rank : 4
Schlafli Type : {6,6,3}
Number of vertices, edges, etc : 18, 54, 27, 3
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,6,3,2} of size 1296
Vertex Figure Of :
{2,6,6,3} of size 1296
{3,6,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,6,3}*216a, {6,6,3}*216b
6-fold quotients : {3,6,3}*108
9-fold quotients : {2,6,3}*72, {6,2,3}*72
18-fold quotients : {3,2,3}*36
27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,6,3}*1296c, {6,6,6}*1296n
3-fold covers : {6,6,9}*1944d, {18,6,3}*1944d, {6,6,3}*1944c, {6,6,3}*1944d, {6,6,3}*1944e, {6,6,3}*1944g
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27);;
s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,12)(13,18)(14,17)(15,16)(19,20)(22,26)
(23,25)(24,27);;
s2 := ( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,16)( 8,18)( 9,17)(19,22)
(20,24)(21,23)(26,27);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)
(16,22)(17,24)(18,23);;
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, s2*s3*s2*s3*s2*s3,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope