# Polytope of Type {2,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9}*648b
if this polytope has a name.
Group : SmallGroup(648,299)
Rank : 4
Schlafli Type : {2,6,9}
Number of vertices, edges, etc : 2, 18, 81, 27
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,6,9,2} of size 1296
Vertex Figure Of :
{2,2,6,9} of size 1296
{3,2,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,6,3}*216
9-fold quotients : {2,6,3}*72
27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,6,9}*1296c, {2,6,18}*1296d
3-fold covers : {2,6,9}*1944a, {2,6,9}*1944b, {2,6,9}*1944c, {2,6,9}*1944d, {2,18,9}*1944j, {6,6,9}*1944g
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 7, 8)(10,11);;
s2 := ( 4, 5)( 6,10)( 7, 9)( 8,11);;
s3 := ( 3, 6)( 4, 8)( 5, 7)(10,11);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(11)!(1,2);
s1 := Sym(11)!( 4, 5)( 7, 8)(10,11);
s2 := Sym(11)!( 4, 5)( 6,10)( 7, 9)( 8,11);
s3 := Sym(11)!( 3, 6)( 4, 8)( 5, 7)(10,11);
poly := sub<Sym(11)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2 >;

```

to this polytope