Polytope of Type {90,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,2}*360
if this polytope has a name.
Group : SmallGroup(360,49)
Rank : 3
Schlafli Type : {90,2}
Number of vertices, edges, etc : 90, 90, 2
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {90,2,2} of size 720
   {90,2,3} of size 1080
   {90,2,4} of size 1440
   {90,2,5} of size 1800
Vertex Figure Of :
   {2,90,2} of size 720
   {4,90,2} of size 1440
   {4,90,2} of size 1440
   {4,90,2} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {45,2}*180
   3-fold quotients : {30,2}*120
   5-fold quotients : {18,2}*72
   6-fold quotients : {15,2}*60
   9-fold quotients : {10,2}*40
   10-fold quotients : {9,2}*36
   15-fold quotients : {6,2}*24
   18-fold quotients : {5,2}*20
   30-fold quotients : {3,2}*12
   45-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {180,2}*720, {90,4}*720a
   3-fold covers : {270,2}*1080, {90,6}*1080a, {90,6}*1080b
   4-fold covers : {180,4}*1440a, {360,2}*1440, {90,8}*1440, {90,4}*1440
   5-fold covers : {450,2}*1800, {90,10}*1800b, {90,10}*1800c
Permutation Representation (GAP) :

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

to this polytope