Part of the Atlas of Small Regular Polytopes

Polytope of Type {90,2}

Atlas Canonical Name {90,2}*360

Overview

Group
SmallGroup(360,49)
Rank
3
Schläfli Type
{90,2}
Vertices, edges, …
90, 90, 2
Order of s0s1s2
90
Order of s0s1s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

3-fold

5-fold

6-fold

9-fold

10-fold

15-fold

18-fold

30-fold

45-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

Representations

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 >;