# Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*972
Also Known As : {3,6}(9,0), {3,6}18if this polytope has another name.
Group : SmallGroup(972,115)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 81, 243, 162
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{3,6,2} of size 1944
Vertex Figure Of :
{2,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,6}*324
9-fold quotients : {3,6}*108
27-fold quotients : {3,6}*36
81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,6}*1944b
Permutation Representation (GAP) :
```s0 := (10,19)(11,20)(12,21)(13,24)(14,22)(15,23)(16,26)(17,27)(18,25);;
s1 := ( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,12)( 8,10)( 9,11);;
s2 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,19)(11,21)(12,20)(13,27)(14,26)(15,25)
(16,22)(17,24)(18,23);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*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)(13,24)(14,22)(15,23)(16,26)(17,27)(18,25);
s1 := Sym(27)!( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,12)( 8,10)( 9,11);
s2 := Sym(27)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,19)(11,21)(12,20)(13,27)(14,26)
(15,25)(16,22)(17,24)(18,23);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

```
References : None.
to this polytope