# Polytope of Type {68,6}

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

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

```
References : None.
to this polytope