# Polytope of Type {3,6,6}

Atlas Canonical Name : {3,6,6}*648c
Also Known As : {{3,6}6,{6,6|2}}. if this polytope has another name.
Group : SmallGroup(648,555)
Rank : 4
Schlafli Type : {3,6,6}
Number of vertices, edges, etc : 9, 27, 54, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
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 : {3,6,2}*216, {3,6,6}*216b
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}*1296e, {6,6,6}*1296c
3-fold covers : {9,6,6}*1944c, {3,6,18}*1944e, {3,6,6}*1944b, {3,6,6}*1944c, {9,6,6}*1944f, {9,6,6}*1944g, {9,6,6}*1944h, {3,6,6}*1944h, {3,18,6}*1944
Permutation Representation (GAP) :
```s0 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27);;
s1 := ( 1,21)( 2,19)( 3,20)( 4,24)( 5,22)( 6,23)( 7,27)( 8,25)( 9,26);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)
(23,27)(24,26);;
s3 := ( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,22)(20,23)(21,24);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope