# Polytope of Type {3,6,6}

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

```
References : None.
to this polytope