Polytope of Type {2,90,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,90,4}*1440a
if this polytope has a name.
Group : SmallGroup(1440,1665)
Rank : 4
Schlafli Type : {2,90,4}
Number of vertices, edges, etc : 2, 90, 180, 4
Order of s₀s₁s₂s₃ : 180
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,90,2}*720
   3-fold quotients : {2,30,4}*480a
   4-fold quotients : {2,45,2}*360
   5-fold quotients : {2,18,4}*288a
   6-fold quotients : {2,30,2}*240
   9-fold quotients : {2,10,4}*160
   10-fold quotients : {2,18,2}*144
   12-fold quotients : {2,15,2}*120
   15-fold quotients : {2,6,4}*96a
   18-fold quotients : {2,10,2}*80
   20-fold quotients : {2,9,2}*72
   30-fold quotients : {2,6,2}*48
   36-fold quotients : {2,5,2}*40
   45-fold quotients : {2,2,4}*32
   60-fold quotients : {2,3,2}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6, 15)(  7, 17)(  8, 16)(  9, 12)( 10, 14)( 11, 13)( 18, 35)
( 19, 34)( 20, 33)( 21, 47)( 22, 46)( 23, 45)( 24, 44)( 25, 43)( 26, 42)
( 27, 41)( 28, 40)( 29, 39)( 30, 38)( 31, 37)( 32, 36)( 49, 50)( 51, 60)
( 52, 62)( 53, 61)( 54, 57)( 55, 59)( 56, 58)( 63, 80)( 64, 79)( 65, 78)
( 66, 92)( 67, 91)( 68, 90)( 69, 89)( 70, 88)( 71, 87)( 72, 86)( 73, 85)
( 74, 84)( 75, 83)( 76, 82)( 77, 81)( 94, 95)( 96,105)( 97,107)( 98,106)
( 99,102)(100,104)(101,103)(108,125)(109,124)(110,123)(111,137)(112,136)
(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)
(121,127)(122,126)(139,140)(141,150)(142,152)(143,151)(144,147)(145,149)
(146,148)(153,170)(154,169)(155,168)(156,182)(157,181)(158,180)(159,179)
(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171);;
s2 := (  3, 21)(  4, 23)(  5, 22)(  6, 18)(  7, 20)(  8, 19)(  9, 30)( 10, 32)
( 11, 31)( 12, 27)( 13, 29)( 14, 28)( 15, 24)( 16, 26)( 17, 25)( 33, 38)
( 34, 37)( 35, 36)( 39, 47)( 40, 46)( 41, 45)( 42, 44)( 48, 66)( 49, 68)
( 50, 67)( 51, 63)( 52, 65)( 53, 64)( 54, 75)( 55, 77)( 56, 76)( 57, 72)
( 58, 74)( 59, 73)( 60, 69)( 61, 71)( 62, 70)( 78, 83)( 79, 82)( 80, 81)
( 84, 92)( 85, 91)( 86, 90)( 87, 89)( 93,156)( 94,158)( 95,157)( 96,153)
( 97,155)( 98,154)( 99,165)(100,167)(101,166)(102,162)(103,164)(104,163)
(105,159)(106,161)(107,160)(108,141)(109,143)(110,142)(111,138)(112,140)
(113,139)(114,150)(115,152)(116,151)(117,147)(118,149)(119,148)(120,144)
(121,146)(122,145)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)
(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,175)
(137,174);;
s3 := (  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)
( 91,181)( 92,182);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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, 35)( 19, 34)( 20, 33)( 21, 47)( 22, 46)( 23, 45)( 24, 44)( 25, 43)
( 26, 42)( 27, 41)( 28, 40)( 29, 39)( 30, 38)( 31, 37)( 32, 36)( 49, 50)
( 51, 60)( 52, 62)( 53, 61)( 54, 57)( 55, 59)( 56, 58)( 63, 80)( 64, 79)
( 65, 78)( 66, 92)( 67, 91)( 68, 90)( 69, 89)( 70, 88)( 71, 87)( 72, 86)
( 73, 85)( 74, 84)( 75, 83)( 76, 82)( 77, 81)( 94, 95)( 96,105)( 97,107)
( 98,106)( 99,102)(100,104)(101,103)(108,125)(109,124)(110,123)(111,137)
(112,136)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)
(120,128)(121,127)(122,126)(139,140)(141,150)(142,152)(143,151)(144,147)
(145,149)(146,148)(153,170)(154,169)(155,168)(156,182)(157,181)(158,180)
(159,179)(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)
(167,171);
s2 := Sym(182)!(  3, 21)(  4, 23)(  5, 22)(  6, 18)(  7, 20)(  8, 19)(  9, 30)
( 10, 32)( 11, 31)( 12, 27)( 13, 29)( 14, 28)( 15, 24)( 16, 26)( 17, 25)
( 33, 38)( 34, 37)( 35, 36)( 39, 47)( 40, 46)( 41, 45)( 42, 44)( 48, 66)
( 49, 68)( 50, 67)( 51, 63)( 52, 65)( 53, 64)( 54, 75)( 55, 77)( 56, 76)
( 57, 72)( 58, 74)( 59, 73)( 60, 69)( 61, 71)( 62, 70)( 78, 83)( 79, 82)
( 80, 81)( 84, 92)( 85, 91)( 86, 90)( 87, 89)( 93,156)( 94,158)( 95,157)
( 96,153)( 97,155)( 98,154)( 99,165)(100,167)(101,166)(102,162)(103,164)
(104,163)(105,159)(106,161)(107,160)(108,141)(109,143)(110,142)(111,138)
(112,140)(113,139)(114,150)(115,152)(116,151)(117,147)(118,149)(119,148)
(120,144)(121,146)(122,145)(123,173)(124,172)(125,171)(126,170)(127,169)
(128,168)(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)
(136,175)(137,174);
s3 := Sym(182)!(  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)( 91,181)( 92,182);
poly := sub<Sym(182)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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