Polytope of Type {2,45}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,45}*180
if this polytope has a name.
Group : SmallGroup(180,11)
Rank : 3
Schlafli Type : {2,45}
Number of vertices, edges, etc : 2, 45, 45
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,45,2} of size 360
   {2,45,4} of size 720
   {2,45,6} of size 1080
   {2,45,4} of size 1440
   {2,45,10} of size 1800
Vertex Figure Of :
   {2,2,45} of size 360
   {3,2,45} of size 540
   {4,2,45} of size 720
   {5,2,45} of size 900
   {6,2,45} of size 1080
   {7,2,45} of size 1260
   {8,2,45} of size 1440
   {9,2,45} of size 1620
   {10,2,45} of size 1800
   {11,2,45} of size 1980
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,15}*60
   5-fold quotients : {2,9}*36
   9-fold quotients : {2,5}*20
   15-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,90}*360
   3-fold covers : {2,135}*540, {6,45}*540
   4-fold covers : {2,180}*720, {4,90}*720a, {4,45}*720
   5-fold covers : {2,225}*900, {10,45}*900
   6-fold covers : {2,270}*1080, {6,90}*1080a, {6,90}*1080b
   7-fold covers : {2,315}*1260
   8-fold covers : {4,180}*1440a, {2,360}*1440, {8,90}*1440, {8,45}*1440, {4,90}*1440
   9-fold covers : {2,405}*1620, {18,45}*1620, {6,45}*1620a, {6,135}*1620
   10-fold covers : {2,450}*1800, {10,90}*1800b, {10,90}*1800c
   11-fold covers : {2,495}*1980
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)
(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)
(46,47);;
s2 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)
(45,46);;
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*s1*s0*s1, s0*s2*s0*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(47)!(1,2);
s1 := Sym(47)!( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)
(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)
(44,45)(46,47);
s2 := Sym(47)!( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)
(43,44)(45,46);
poly := sub<Sym(47)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s1*s0*s1, s0*s2*s0*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 >;

to this polytope