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Polytope of Type {4,90}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,90}*720a
Also Known As : {4,90|2}. if this polytope has another name.
Group : SmallGroup(720,179)
Rank : 3
Schlafli Type : {4,90}
Number of vertices, edges, etc : 4, 180, 90
Order of s0s1s2 : 180
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,90,2} of size 1440
Vertex Figure Of :
   {2,4,90} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,90}*360
   3-fold quotients : {4,30}*240a
   4-fold quotients : {2,45}*180
   5-fold quotients : {4,18}*144a
   6-fold quotients : {2,30}*120
   9-fold quotients : {4,10}*80
   10-fold quotients : {2,18}*72
   12-fold quotients : {2,15}*60
   15-fold quotients : {4,6}*48a
   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 : {4,2}*16
   60-fold quotients : {2,3}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,180}*1440a, {8,90}*1440
Permutation Representation (GAP) :
s0 := ( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)
( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,151)
(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)
(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)
(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)
(131,176)(132,177)(133,178)(134,179)(135,180);;
s1 := (  1, 91)(  2, 93)(  3, 92)(  4,103)(  5,105)(  6,104)(  7,100)(  8,102)
(  9,101)( 10, 97)( 11, 99)( 12, 98)( 13, 94)( 14, 96)( 15, 95)( 16,123)
( 17,122)( 18,121)( 19,135)( 20,134)( 21,133)( 22,132)( 23,131)( 24,130)
( 25,129)( 26,128)( 27,127)( 28,126)( 29,125)( 30,124)( 31,108)( 32,107)
( 33,106)( 34,120)( 35,119)( 36,118)( 37,117)( 38,116)( 39,115)( 40,114)
( 41,113)( 42,112)( 43,111)( 44,110)( 45,109)( 46,136)( 47,138)( 48,137)
( 49,148)( 50,150)( 51,149)( 52,145)( 53,147)( 54,146)( 55,142)( 56,144)
( 57,143)( 58,139)( 59,141)( 60,140)( 61,168)( 62,167)( 63,166)( 64,180)
( 65,179)( 66,178)( 67,177)( 68,176)( 69,175)( 70,174)( 71,173)( 72,172)
( 73,171)( 74,170)( 75,169)( 76,153)( 77,152)( 78,151)( 79,165)( 80,164)
( 81,163)( 82,162)( 83,161)( 84,160)( 85,159)( 86,158)( 87,157)( 88,156)
( 89,155)( 90,154);;
s2 := (  1, 19)(  2, 21)(  3, 20)(  4, 16)(  5, 18)(  6, 17)(  7, 28)(  8, 30)
(  9, 29)( 10, 25)( 11, 27)( 12, 26)( 13, 22)( 14, 24)( 15, 23)( 31, 36)
( 32, 35)( 33, 34)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 64)( 47, 66)
( 48, 65)( 49, 61)( 50, 63)( 51, 62)( 52, 73)( 53, 75)( 54, 74)( 55, 70)
( 56, 72)( 57, 71)( 58, 67)( 59, 69)( 60, 68)( 76, 81)( 77, 80)( 78, 79)
( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,109)( 92,111)( 93,110)( 94,106)
( 95,108)( 96,107)( 97,118)( 98,120)( 99,119)(100,115)(101,117)(102,116)
(103,112)(104,114)(105,113)(121,126)(122,125)(123,124)(127,135)(128,134)
(129,133)(130,132)(136,154)(137,156)(138,155)(139,151)(140,153)(141,152)
(142,163)(143,165)(144,164)(145,160)(146,162)(147,161)(148,157)(149,159)
(150,158)(166,171)(167,170)(168,169)(172,180)(173,179)(174,178)(175,177);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(180)!( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)
( 98,143)( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)
(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)
(122,167)(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)
(130,175)(131,176)(132,177)(133,178)(134,179)(135,180);
s1 := Sym(180)!(  1, 91)(  2, 93)(  3, 92)(  4,103)(  5,105)(  6,104)(  7,100)
(  8,102)(  9,101)( 10, 97)( 11, 99)( 12, 98)( 13, 94)( 14, 96)( 15, 95)
( 16,123)( 17,122)( 18,121)( 19,135)( 20,134)( 21,133)( 22,132)( 23,131)
( 24,130)( 25,129)( 26,128)( 27,127)( 28,126)( 29,125)( 30,124)( 31,108)
( 32,107)( 33,106)( 34,120)( 35,119)( 36,118)( 37,117)( 38,116)( 39,115)
( 40,114)( 41,113)( 42,112)( 43,111)( 44,110)( 45,109)( 46,136)( 47,138)
( 48,137)( 49,148)( 50,150)( 51,149)( 52,145)( 53,147)( 54,146)( 55,142)
( 56,144)( 57,143)( 58,139)( 59,141)( 60,140)( 61,168)( 62,167)( 63,166)
( 64,180)( 65,179)( 66,178)( 67,177)( 68,176)( 69,175)( 70,174)( 71,173)
( 72,172)( 73,171)( 74,170)( 75,169)( 76,153)( 77,152)( 78,151)( 79,165)
( 80,164)( 81,163)( 82,162)( 83,161)( 84,160)( 85,159)( 86,158)( 87,157)
( 88,156)( 89,155)( 90,154);
s2 := Sym(180)!(  1, 19)(  2, 21)(  3, 20)(  4, 16)(  5, 18)(  6, 17)(  7, 28)
(  8, 30)(  9, 29)( 10, 25)( 11, 27)( 12, 26)( 13, 22)( 14, 24)( 15, 23)
( 31, 36)( 32, 35)( 33, 34)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 64)
( 47, 66)( 48, 65)( 49, 61)( 50, 63)( 51, 62)( 52, 73)( 53, 75)( 54, 74)
( 55, 70)( 56, 72)( 57, 71)( 58, 67)( 59, 69)( 60, 68)( 76, 81)( 77, 80)
( 78, 79)( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,109)( 92,111)( 93,110)
( 94,106)( 95,108)( 96,107)( 97,118)( 98,120)( 99,119)(100,115)(101,117)
(102,116)(103,112)(104,114)(105,113)(121,126)(122,125)(123,124)(127,135)
(128,134)(129,133)(130,132)(136,154)(137,156)(138,155)(139,151)(140,153)
(141,152)(142,163)(143,165)(144,164)(145,160)(146,162)(147,161)(148,157)
(149,159)(150,158)(166,171)(167,170)(168,169)(172,180)(173,179)(174,178)
(175,177);
poly := sub<Sym(180)|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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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