Polytope of Type {90}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90}*180
Also Known As : 90-gon, {90}. if this polytope has another name.
Group : SmallGroup(180,11)
Rank : 2
Schlafli Type : {90}
Number of vertices, edges, etc : 90, 90
Order of s0s1 : 90
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {90,2} of size 360
   {90,4} of size 720
   {90,4} of size 720
   {90,4} of size 720
   {90,6} of size 1080
   {90,6} of size 1080
   {90,8} of size 1440
   {90,4} of size 1440
   {90,6} of size 1620
   {90,6} of size 1620
   {90,6} of size 1620
   {90,10} of size 1800
   {90,10} of size 1800
   {90,10} of size 1800
Vertex Figure Of :
   {2,90} of size 360
   {4,90} of size 720
   {4,90} of size 720
   {4,90} of size 720
   {6,90} of size 1080
   {6,90} of size 1080
   {8,90} of size 1440
   {4,90} of size 1440
   {6,90} of size 1620
   {6,90} of size 1620
   {6,90} of size 1620
   {10,90} of size 1800
   {10,90} of size 1800
   {10,90} of size 1800
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {45}*90
   3-fold quotients : {30}*60
   5-fold quotients : {18}*36
   6-fold quotients : {15}*30
   9-fold quotients : {10}*20
   10-fold quotients : {9}*18
   15-fold quotients : {6}*12
   18-fold quotients : {5}*10
   30-fold quotients : {3}*6
   45-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {180}*360
   3-fold covers : {270}*540
   4-fold covers : {360}*720
   5-fold covers : {450}*900
   6-fold covers : {540}*1080
   7-fold covers : {630}*1260
   8-fold covers : {720}*1440
   9-fold covers : {810}*1620
   10-fold covers : {900}*1800
   11-fold covers : {990}*1980
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);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(90)!( 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(90)!( 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);
poly := sub<Sym(90)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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.
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