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# Polytope of Type {6,15}

Atlas Canonical Name : {6,15}*720e
if this polytope has a name.
Group : SmallGroup(720,794)
Rank : 3
Schlafli Type : {6,15}
Number of vertices, edges, etc : 24, 180, 60
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,15,2} of size 1440
Vertex Figure Of :
{2,6,15} of size 1440
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,15}*240
4-fold quotients : {6,15}*180
5-fold quotients : {6,3}*144
12-fold quotients : {2,15}*60
15-fold quotients : {6,3}*48
20-fold quotients : {6,3}*36
30-fold quotients : {3,3}*24
36-fold quotients : {2,5}*20
60-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,15}*1440c, {6,30}*1440h
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27)(30,31)(34,35)(38,39)
(42,43)(46,47)(50,51)(54,55)(58,59);;
s1 := ( 2, 4)( 5,17)( 6,20)( 7,19)( 8,18)( 9,13)(10,16)(11,15)(12,14)(21,41)
(22,44)(23,43)(24,42)(25,57)(26,60)(27,59)(28,58)(29,53)(30,56)(31,55)(32,54)
(33,49)(34,52)(35,51)(36,50)(37,45)(38,48)(39,47)(40,46);;
s2 := ( 1,28)( 2,26)( 3,27)( 4,25)( 5,24)( 6,22)( 7,23)( 8,21)( 9,40)(10,38)
(11,39)(12,37)(13,36)(14,34)(15,35)(16,33)(17,32)(18,30)(19,31)(20,29)(41,48)
(42,46)(43,47)(44,45)(49,60)(50,58)(51,59)(52,57)(53,56);;
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,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(60)!( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27)(30,31)(34,35)
(38,39)(42,43)(46,47)(50,51)(54,55)(58,59);
s1 := Sym(60)!( 2, 4)( 5,17)( 6,20)( 7,19)( 8,18)( 9,13)(10,16)(11,15)(12,14)
(21,41)(22,44)(23,43)(24,42)(25,57)(26,60)(27,59)(28,58)(29,53)(30,56)(31,55)
(32,54)(33,49)(34,52)(35,51)(36,50)(37,45)(38,48)(39,47)(40,46);
s2 := Sym(60)!( 1,28)( 2,26)( 3,27)( 4,25)( 5,24)( 6,22)( 7,23)( 8,21)( 9,40)
(10,38)(11,39)(12,37)(13,36)(14,34)(15,35)(16,33)(17,32)(18,30)(19,31)(20,29)
(41,48)(42,46)(43,47)(44,45)(49,60)(50,58)(51,59)(52,57)(53,56);
poly := sub<Sym(60)|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,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope