Polytope of Type {45}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {45}*90
Also Known As : 45-gon, {45}. if this polytope has another name.
Group : SmallGroup(90,3)
Rank : 2
Schlafli Type : {45}
Number of vertices, edges, etc : 45, 45
Order of s₀s₁ : 45
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {45,2} of size 180
   {45,4} of size 360
   {45,6} of size 540
   {45,4} of size 720
   {45,10} of size 900
   {45,6} of size 1080
   {45,10} of size 1080
   {45,8} of size 1440
   {45,4} of size 1440
   {45,18} of size 1620
   {45,6} of size 1620
   {45,6} of size 1620
   {45,6} of size 1620
   {45,6} of size 1620
Vertex Figure Of :
   {2,45} of size 180
   {4,45} of size 360
   {6,45} of size 540
   {4,45} of size 720
   {10,45} of size 900
   {6,45} of size 1080
   {10,45} of size 1080
   {8,45} of size 1440
   {4,45} of size 1440
   {18,45} of size 1620
   {6,45} of size 1620
   {6,45} of size 1620
   {6,45} of size 1620
   {6,45} of size 1620
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {15}*30
   5-fold quotients : {9}*18
   9-fold quotients : {5}*10
   15-fold quotients : {3}*6
Covers (Minimal Covers in Boldface) :
   2-fold covers : {90}*180
   3-fold covers : {135}*270
   4-fold covers : {180}*360
   5-fold covers : {225}*450
   6-fold covers : {270}*540
   7-fold covers : {315}*630
   8-fold covers : {360}*720
   9-fold covers : {405}*810
   10-fold covers : {450}*900
   11-fold covers : {495}*990
   12-fold covers : {540}*1080
   13-fold covers : {585}*1170
   14-fold covers : {630}*1260
   15-fold covers : {675}*1350
   16-fold covers : {720}*1440
   17-fold covers : {765}*1530
   18-fold covers : {810}*1620
   19-fold covers : {855}*1710
   20-fold covers : {900}*1800
   21-fold covers : {945}*1890
   22-fold covers : {990}*1980
Permutation Representation (GAP) :

s0 := ( 2, 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);;
s1 := ( 1, 2)( 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);;
poly := Group([s0,s1]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(45)!( 2, 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);
s1 := Sym(45)!( 1, 2)( 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);
poly := sub<Sym(45)|s0,s1>;

Finitely Presented Group Representation (Magma) :

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