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# Polytope of Type {6,9}

Atlas Canonical Name : {6,9}*324b
if this polytope has a name.
Group : SmallGroup(324,39)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 18, 81, 27
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,9,2} of size 648
{6,9,4} of size 1296
{6,9,6} of size 1944
Vertex Figure Of :
{2,6,9} of size 648
{4,6,9} of size 1296
{6,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,3}*108
9-fold quotients : {6,3}*36
27-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,18}*648d
3-fold covers : {6,9}*972a, {6,9}*972b, {6,9}*972c, {6,9}*972d, {18,9}*972j
4-fold covers : {6,36}*1296c, {12,18}*1296g, {6,9}*1296c, {12,9}*1296d
5-fold covers : {6,45}*1620c
6-fold covers : {6,18}*1944a, {6,18}*1944d, {6,18}*1944f, {6,18}*1944h, {18,18}*1944ac, {6,18}*1944q
Permutation Representation (GAP) :
```s0 := (2,3)(5,6)(8,9);;
s1 := (2,3)(4,8)(5,7)(6,9);;
s2 := (1,4)(2,6)(3,5)(8,9);;
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,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope