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# Polytope of Type {9,12}

Atlas Canonical Name : {9,12}*648b
if this polytope has a name.
Group : SmallGroup(648,703)
Rank : 3
Schlafli Type : {9,12}
Number of vertices, edges, etc : 27, 162, 36
Order of s0s1s2 : 3
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{9,12,2} of size 1296
Vertex Figure Of :
{2,9,12} of size 1296
Quotients (Maximal Quotients in Boldface) :
27-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {9,12}*1296f, {18,12}*1296n, {18,12}*1296p
3-fold covers : {9,12}*1944c
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4,19)( 5,21)( 6,20)( 7,10)( 8,12)( 9,11)(13,25)(14,27)(15,26)
(17,18)(23,24);;
s1 := ( 1, 4)( 2,22)( 3,13)( 5,19)( 6,10)( 8,25)( 9,16)(11,24)(12,15)(14,21)
(17,27)(20,23);;
s2 := ( 1,16)( 2,17)( 3,18)( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)(19,25)
(20,26)(21,27);;
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*s2*s0*s1*s2*s0*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope