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# Polytope of Type {51,6}

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

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

```
References : None.
to this polytope