Polytope of Type {2,180}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,180}*720
if this polytope has a name.
Group : SmallGroup(720,177)
Rank : 3
Schlafli Type : {2,180}
Number of vertices, edges, etc : 2, 180, 180
Order of s₀s₁s₂ : 180
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,180,2} of size 1440
Vertex Figure Of :
   {2,2,180} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,90}*360
   3-fold quotients : {2,60}*240
   4-fold quotients : {2,45}*180
   5-fold quotients : {2,36}*144
   6-fold quotients : {2,30}*120
   9-fold quotients : {2,20}*80
   10-fold quotients : {2,18}*72
   12-fold quotients : {2,15}*60
   15-fold quotients : {2,12}*48
   18-fold quotients : {2,10}*40
   20-fold quotients : {2,9}*36
   30-fold quotients : {2,6}*24
   36-fold quotients : {2,5}*20
   45-fold quotients : {2,4}*16
   60-fold quotients : {2,3}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,180}*1440a, {2,360}*1440
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6, 15)(  7, 17)(  8, 16)(  9, 12)( 10, 14)( 11, 13)( 18, 34)
( 19, 33)( 20, 35)( 21, 46)( 22, 45)( 23, 47)( 24, 43)( 25, 42)( 26, 44)
( 27, 40)( 28, 39)( 29, 41)( 30, 37)( 31, 36)( 32, 38)( 49, 50)( 51, 60)
( 52, 62)( 53, 61)( 54, 57)( 55, 59)( 56, 58)( 63, 79)( 64, 78)( 65, 80)
( 66, 91)( 67, 90)( 68, 92)( 69, 88)( 70, 87)( 71, 89)( 72, 85)( 73, 84)
( 74, 86)( 75, 82)( 76, 81)( 77, 83)( 93,138)( 94,140)( 95,139)( 96,150)
( 97,152)( 98,151)( 99,147)(100,149)(101,148)(102,144)(103,146)(104,145)
(105,141)(106,143)(107,142)(108,169)(109,168)(110,170)(111,181)(112,180)
(113,182)(114,178)(115,177)(116,179)(117,175)(118,174)(119,176)(120,172)
(121,171)(122,173)(123,154)(124,153)(125,155)(126,166)(127,165)(128,167)
(129,163)(130,162)(131,164)(132,160)(133,159)(134,161)(135,157)(136,156)
(137,158);;
s2 := (  3,111)(  4,113)(  5,112)(  6,108)(  7,110)(  8,109)(  9,120)( 10,122)
( 11,121)( 12,117)( 13,119)( 14,118)( 15,114)( 16,116)( 17,115)( 18, 96)
( 19, 98)( 20, 97)( 21, 93)( 22, 95)( 23, 94)( 24,105)( 25,107)( 26,106)
( 27,102)( 28,104)( 29,103)( 30, 99)( 31,101)( 32,100)( 33,127)( 34,126)
( 35,128)( 36,124)( 37,123)( 38,125)( 39,136)( 40,135)( 41,137)( 42,133)
( 43,132)( 44,134)( 45,130)( 46,129)( 47,131)( 48,156)( 49,158)( 50,157)
( 51,153)( 52,155)( 53,154)( 54,165)( 55,167)( 56,166)( 57,162)( 58,164)
( 59,163)( 60,159)( 61,161)( 62,160)( 63,141)( 64,143)( 65,142)( 66,138)
( 67,140)( 68,139)( 69,150)( 70,152)( 71,151)( 72,147)( 73,149)( 74,148)
( 75,144)( 76,146)( 77,145)( 78,172)( 79,171)( 80,173)( 81,169)( 82,168)
( 83,170)( 84,181)( 85,180)( 86,182)( 87,178)( 88,177)( 89,179)( 90,175)
( 91,174)( 92,176);;
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*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*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*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(182)!(1,2);
s1 := Sym(182)!(  4,  5)(  6, 15)(  7, 17)(  8, 16)(  9, 12)( 10, 14)( 11, 13)
( 18, 34)( 19, 33)( 20, 35)( 21, 46)( 22, 45)( 23, 47)( 24, 43)( 25, 42)
( 26, 44)( 27, 40)( 28, 39)( 29, 41)( 30, 37)( 31, 36)( 32, 38)( 49, 50)
( 51, 60)( 52, 62)( 53, 61)( 54, 57)( 55, 59)( 56, 58)( 63, 79)( 64, 78)
( 65, 80)( 66, 91)( 67, 90)( 68, 92)( 69, 88)( 70, 87)( 71, 89)( 72, 85)
( 73, 84)( 74, 86)( 75, 82)( 76, 81)( 77, 83)( 93,138)( 94,140)( 95,139)
( 96,150)( 97,152)( 98,151)( 99,147)(100,149)(101,148)(102,144)(103,146)
(104,145)(105,141)(106,143)(107,142)(108,169)(109,168)(110,170)(111,181)
(112,180)(113,182)(114,178)(115,177)(116,179)(117,175)(118,174)(119,176)
(120,172)(121,171)(122,173)(123,154)(124,153)(125,155)(126,166)(127,165)
(128,167)(129,163)(130,162)(131,164)(132,160)(133,159)(134,161)(135,157)
(136,156)(137,158);
s2 := Sym(182)!(  3,111)(  4,113)(  5,112)(  6,108)(  7,110)(  8,109)(  9,120)
( 10,122)( 11,121)( 12,117)( 13,119)( 14,118)( 15,114)( 16,116)( 17,115)
( 18, 96)( 19, 98)( 20, 97)( 21, 93)( 22, 95)( 23, 94)( 24,105)( 25,107)
( 26,106)( 27,102)( 28,104)( 29,103)( 30, 99)( 31,101)( 32,100)( 33,127)
( 34,126)( 35,128)( 36,124)( 37,123)( 38,125)( 39,136)( 40,135)( 41,137)
( 42,133)( 43,132)( 44,134)( 45,130)( 46,129)( 47,131)( 48,156)( 49,158)
( 50,157)( 51,153)( 52,155)( 53,154)( 54,165)( 55,167)( 56,166)( 57,162)
( 58,164)( 59,163)( 60,159)( 61,161)( 62,160)( 63,141)( 64,143)( 65,142)
( 66,138)( 67,140)( 68,139)( 69,150)( 70,152)( 71,151)( 72,147)( 73,149)
( 74,148)( 75,144)( 76,146)( 77,145)( 78,172)( 79,171)( 80,173)( 81,169)
( 82,168)( 83,170)( 84,181)( 85,180)( 86,182)( 87,178)( 88,177)( 89,179)
( 90,175)( 91,174)( 92,176);
poly := sub<Sym(182)|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*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*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*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