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# Polytope of Type {69,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {69,4}*552
if this polytope has a name.
Group : SmallGroup(552,32)
Rank : 3
Schlafli Type : {69,4}
Number of vertices, edges, etc : 69, 138, 4
Order of s0s1s2 : 69
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{69,4,2} of size 1104
Vertex Figure Of :
{2,69,4} of size 1104
Quotients (Maximal Quotients in Boldface) :
23-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {69,4}*1104, {138,4}*1104b, {138,4}*1104c
3-fold covers : {207,4}*1656
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5,89)( 6,91)( 7,90)( 8,92)( 9,85)(10,87)(11,86)(12,88)(13,81)
(14,83)(15,82)(16,84)(17,77)(18,79)(19,78)(20,80)(21,73)(22,75)(23,74)(24,76)
(25,69)(26,71)(27,70)(28,72)(29,65)(30,67)(31,66)(32,68)(33,61)(34,63)(35,62)
(36,64)(37,57)(38,59)(39,58)(40,60)(41,53)(42,55)(43,54)(44,56)(45,49)(46,51)
(47,50)(48,52);;
s1 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,89)(10,90)(11,92)(12,91)(13,85)(14,86)
(15,88)(16,87)(17,81)(18,82)(19,84)(20,83)(21,77)(22,78)(23,80)(24,79)(25,73)
(26,74)(27,76)(28,75)(29,69)(30,70)(31,72)(32,71)(33,65)(34,66)(35,68)(36,67)
(37,61)(38,62)(39,64)(40,63)(41,57)(42,58)(43,60)(44,59)(45,53)(46,54)(47,56)
(48,55)(51,52);;
s2 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)
(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)
(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)
(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79)(81,84)(82,83)(85,88)
(86,87)(89,92)(90,91);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s1*s0*s2*s1*s2*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope