Polytope of Type {90,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,4}*720a
Also Known As : {90,4|2}. if this polytope has another name.
Group : SmallGroup(720,179)
Rank : 3
Schlafli Type : {90,4}
Number of vertices, edges, etc : 90, 180, 4
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 :
   {90,4,2} of size 1440
Vertex Figure Of :
   {2,90,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {90,2}*360
   3-fold quotients : {30,4}*240a
   4-fold quotients : {45,2}*180
   5-fold quotients : {18,4}*144a
   6-fold quotients : {30,2}*120
   9-fold quotients : {10,4}*80
   10-fold quotients : {18,2}*72
   12-fold quotients : {15,2}*60
   15-fold quotients : {6,4}*48a
   18-fold quotients : {10,2}*40
   20-fold quotients : {9,2}*36
   30-fold quotients : {6,2}*24
   36-fold quotients : {5,2}*20
   45-fold quotients : {2,4}*16
   60-fold quotients : {3,2}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {180,4}*1440a, {90,8}*1440
Permutation Representation (GAP) :

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