Polytope of Type {4,69}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,69}*552
if this polytope has a name.
Group : SmallGroup(552,32)
Rank : 3
Schlafli Type : {4,69}
Number of vertices, edges, etc : 4, 138, 69
Order of s₀s₁s₂ : 69
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,69,2} of size 1104
Vertex Figure Of :
   {2,4,69} of size 1104
Quotients (Maximal Quotients in Boldface) :
   23-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,69}*1104, {4,138}*1104b, {4,138}*1104c
   3-fold covers : {4,207}*1656
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope