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# Polytope of Type {6,66}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,66}*792a
if this polytope has a name.
Group : SmallGroup(792,116)
Rank : 3
Schlafli Type : {6,66}
Number of vertices, edges, etc : 6, 198, 66
Order of s0s1s2 : 66
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,66,2} of size 1584
Vertex Figure Of :
{2,6,66} of size 1584
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,22}*264
9-fold quotients : {2,22}*88
11-fold quotients : {6,6}*72c
18-fold quotients : {2,11}*44
22-fold quotients : {3,6}*36
33-fold quotients : {6,2}*24
66-fold quotients : {3,2}*12
99-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,132}*1584a, {12,66}*1584a
Permutation Representation (GAP) :
```s0 := (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)
(22,33)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)
(44,77)(45,89)(46,90)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)
(55,99)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)
(66,88);;
s1 := ( 1,45)( 2,55)( 3,54)( 4,53)( 5,52)( 6,51)( 7,50)( 8,49)( 9,48)(10,47)
(11,46)(12,34)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)
(22,35)(23,56)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)(32,58)
(33,57)(67,78)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)
(77,79)(90,99)(91,98)(92,97)(93,96)(94,95);;
s2 := ( 1, 2)( 3,11)( 4,10)( 5, 9)( 6, 8)(12,24)(13,23)(14,33)(15,32)(16,31)
(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(34,35)(36,44)(37,43)(38,42)(39,41)
(45,57)(46,56)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)
(67,68)(69,77)(70,76)(71,75)(72,74)(78,90)(79,89)(80,99)(81,98)(82,97)(83,96)
(84,95)(85,94)(86,93)(87,92)(88,91);;
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,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(99)!(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)
(21,32)(22,33)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)
(43,76)(44,77)(45,89)(46,90)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)
(54,98)(55,99)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)
(65,87)(66,88);
s1 := Sym(99)!( 1,45)( 2,55)( 3,54)( 4,53)( 5,52)( 6,51)( 7,50)( 8,49)( 9,48)
(10,47)(11,46)(12,34)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)
(21,36)(22,35)(23,56)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)
(32,58)(33,57)(67,78)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)
(76,80)(77,79)(90,99)(91,98)(92,97)(93,96)(94,95);
s2 := Sym(99)!( 1, 2)( 3,11)( 4,10)( 5, 9)( 6, 8)(12,24)(13,23)(14,33)(15,32)
(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(34,35)(36,44)(37,43)(38,42)
(39,41)(45,57)(46,56)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)
(55,58)(67,68)(69,77)(70,76)(71,75)(72,74)(78,90)(79,89)(80,99)(81,98)(82,97)
(83,96)(84,95)(85,94)(86,93)(87,92)(88,91);
poly := sub<Sym(99)|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,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 >;

```
References : None.
to this polytope