Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*648f
if this polytope has a name.
Group : SmallGroup(648,555)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 54, 162, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,6,2} of size 1296
Vertex Figure Of :
{2,6,6} of size 1296
{3,6,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,6}*216a, {6,6}*216d
6-fold quotients : {6,3}*108
9-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
18-fold quotients : {3,6}*36, {6,3}*36
27-fold quotients : {2,6}*24, {6,2}*24
54-fold quotients : {2,3}*12, {3,2}*12
81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*1296h, {12,6}*1296i
3-fold covers : {6,18}*1944m, {18,6}*1944o, {6,6}*1944d, {6,6}*1944f, {6,18}*1944p, {6,18}*1944q, {6,18}*1944r, {6,6}*1944i, {18,6}*1944u
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)
(23,27)(24,26);;
s1 := ( 1, 4)( 2, 5)( 3, 6)(10,23)(11,24)(12,22)(13,20)(14,21)(15,19)(16,26)
(17,27)(18,25);;
s2 := ( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope