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# Polytope of Type {6,51}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,51}*816
if this polytope has a name.
Group : SmallGroup(816,190)
Rank : 3
Schlafli Type : {6,51}
Number of vertices, edges, etc : 8, 204, 68
Order of s0s1s2 : 68
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,51,2} of size 1632
Vertex Figure Of :
{2,6,51} of size 1632
Quotients (Maximal Quotients in Boldface) :
12-fold quotients : {2,17}*68
17-fold quotients : {6,3}*48
34-fold quotients : {3,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,51}*1632, {6,102}*1632
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)(62,63)(66,67);;
s1 := ( 3, 4)( 5,65)( 6,66)( 7,68)( 8,67)( 9,61)(10,62)(11,64)(12,63)(13,57)
(14,58)(15,60)(16,59)(17,53)(18,54)(19,56)(20,55)(21,49)(22,50)(23,52)(24,51)
(25,45)(26,46)(27,48)(28,47)(29,41)(30,42)(31,44)(32,43)(33,37)(34,38)(35,40)
(36,39);;
s2 := ( 1, 8)( 2, 6)( 3, 7)( 4, 5)( 9,68)(10,66)(11,67)(12,65)(13,64)(14,62)
(15,63)(16,61)(17,60)(18,58)(19,59)(20,57)(21,56)(22,54)(23,55)(24,53)(25,52)
(26,50)(27,51)(28,49)(29,48)(30,46)(31,47)(32,45)(33,44)(34,42)(35,43)(36,41)
(37,40);;
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*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(68)!( 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)(62,63)(66,67);
s1 := Sym(68)!( 3, 4)( 5,65)( 6,66)( 7,68)( 8,67)( 9,61)(10,62)(11,64)(12,63)
(13,57)(14,58)(15,60)(16,59)(17,53)(18,54)(19,56)(20,55)(21,49)(22,50)(23,52)
(24,51)(25,45)(26,46)(27,48)(28,47)(29,41)(30,42)(31,44)(32,43)(33,37)(34,38)
(35,40)(36,39);
s2 := Sym(68)!( 1, 8)( 2, 6)( 3, 7)( 4, 5)( 9,68)(10,66)(11,67)(12,65)(13,64)
(14,62)(15,63)(16,61)(17,60)(18,58)(19,59)(20,57)(21,56)(22,54)(23,55)(24,53)
(25,52)(26,50)(27,51)(28,49)(29,48)(30,46)(31,47)(32,45)(33,44)(34,42)(35,43)
(36,41)(37,40);
poly := sub<Sym(68)|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*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2 >;

```
References : None.
to this polytope