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# Polytope of Type {12,6}

Atlas Canonical Name : {12,6}*864c
if this polytope has a name.
Group : SmallGroup(864,4673)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 72, 216, 36
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 :
{12,6,2} of size 1728
Vertex Figure Of :
{2,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {12,6}*288a, {12,6}*288b
4-fold quotients : {6,6}*216d
6-fold quotients : {12,3}*144, {12,6}*144d
9-fold quotients : {4,6}*96
12-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
18-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {3,6}*36, {6,3}*36
36-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
72-fold quotients : {2,3}*12, {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,6}*1728f, {24,6}*1728g, {12,12}*1728w, {12,6}*1728i, {12,12}*1728y
Permutation Representation (GAP) :
```s0 := ( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,27)(14,28)(15,25)(16,26)
(17,35)(18,36)(19,33)(20,34)(21,31)(22,32)(23,29)(24,30);;
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)(25,29)(26,31)(27,30)(28,32)(34,35);;
s2 := ( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(14,16)(17,21)(18,24)(19,23)(20,22)
(26,28)(29,33)(30,36)(31,35)(32,34);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*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,
s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope