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# Polytope of Type {9,12}

Atlas Canonical Name : {9,12}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1788)
Rank : 3
Schlafli Type : {9,12}
Number of vertices, edges, etc : 54, 324, 72
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,12}*432
4-fold quotients : {9,6}*324c
9-fold quotients : {3,12}*144
12-fold quotients : {3,6}*108
27-fold quotients : {3,4}*48
36-fold quotients : {3,6}*36
54-fold quotients : {3,4}*24
108-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)
(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);;
s1 := ( 1,17)( 2,19)( 3,18)( 4,20)( 5,13)( 6,15)( 7,14)( 8,16)( 9,21)(10,23)
(11,22)(12,24)(25,33)(26,35)(27,34)(28,36)(30,31);;
s2 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,26)(14,25)(15,28)(16,27)
(17,30)(18,29)(19,32)(20,31)(21,34)(22,33)(23,36)(24,35);;
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*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)
(17,33)(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);
s1 := Sym(36)!( 1,17)( 2,19)( 3,18)( 4,20)( 5,13)( 6,15)( 7,14)( 8,16)( 9,21)
(10,23)(11,22)(12,24)(25,33)(26,35)(27,34)(28,36)(30,31);
s2 := Sym(36)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,26)(14,25)(15,28)
(16,27)(17,30)(18,29)(19,32)(20,31)(21,34)(22,33)(23,36)(24,35);
poly := sub<Sym(36)|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*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope