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Polytope of Type {2,20,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,20,18}*1440a
if this polytope has a name.
Group : SmallGroup(1440,1584)
Rank : 4
Schlafli Type : {2,20,18}
Number of vertices, edges, etc : 2, 20, 180, 18
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 : {2,10,18}*720
   3-fold quotients : {2,20,6}*480a
   5-fold quotients : {2,4,18}*288a
   6-fold quotients : {2,10,6}*240
   9-fold quotients : {2,20,2}*160
   10-fold quotients : {2,2,18}*144
   15-fold quotients : {2,4,6}*96a
   18-fold quotients : {2,10,2}*80
   20-fold quotients : {2,2,9}*72
   30-fold quotients : {2,2,6}*48
   36-fold quotients : {2,5,2}*40
   45-fold quotients : {2,4,2}*32
   60-fold quotients : {2,2,3}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  6, 15)(  7, 16)(  8, 17)(  9, 12)( 10, 13)( 11, 14)( 21, 30)( 22, 31)
( 23, 32)( 24, 27)( 25, 28)( 26, 29)( 36, 45)( 37, 46)( 38, 47)( 39, 42)
( 40, 43)( 41, 44)( 51, 60)( 52, 61)( 53, 62)( 54, 57)( 55, 58)( 56, 59)
( 66, 75)( 67, 76)( 68, 77)( 69, 72)( 70, 73)( 71, 74)( 81, 90)( 82, 91)
( 83, 92)( 84, 87)( 85, 88)( 86, 89)( 93,138)( 94,139)( 95,140)( 96,150)
( 97,151)( 98,152)( 99,147)(100,148)(101,149)(102,144)(103,145)(104,146)
(105,141)(106,142)(107,143)(108,153)(109,154)(110,155)(111,165)(112,166)
(113,167)(114,162)(115,163)(116,164)(117,159)(118,160)(119,161)(120,156)
(121,157)(122,158)(123,168)(124,169)(125,170)(126,180)(127,181)(128,182)
(129,177)(130,178)(131,179)(132,174)(133,175)(134,176)(135,171)(136,172)
(137,173);;
s2 := (  3, 96)(  4, 98)(  5, 97)(  6, 93)(  7, 95)(  8, 94)(  9,105)( 10,107)
( 11,106)( 12,102)( 13,104)( 14,103)( 15, 99)( 16,101)( 17,100)( 18,128)
( 19,127)( 20,126)( 21,125)( 22,124)( 23,123)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30,131)( 31,130)( 32,129)( 33,113)( 34,112)
( 35,111)( 36,110)( 37,109)( 38,108)( 39,122)( 40,121)( 41,120)( 42,119)
( 43,118)( 44,117)( 45,116)( 46,115)( 47,114)( 48,141)( 49,143)( 50,142)
( 51,138)( 52,140)( 53,139)( 54,150)( 55,152)( 56,151)( 57,147)( 58,149)
( 59,148)( 60,144)( 61,146)( 62,145)( 63,173)( 64,172)( 65,171)( 66,170)
( 67,169)( 68,168)( 69,182)( 70,181)( 71,180)( 72,179)( 73,178)( 74,177)
( 75,176)( 76,175)( 77,174)( 78,158)( 79,157)( 80,156)( 81,155)( 82,154)
( 83,153)( 84,167)( 85,166)( 86,165)( 87,164)( 88,163)( 89,162)( 90,161)
( 91,160)( 92,159);;
s3 := (  3, 18)(  4, 20)(  5, 19)(  6, 21)(  7, 23)(  8, 22)(  9, 24)( 10, 26)
( 11, 25)( 12, 27)( 13, 29)( 14, 28)( 15, 30)( 16, 32)( 17, 31)( 33, 35)
( 36, 38)( 39, 41)( 42, 44)( 45, 47)( 48, 63)( 49, 65)( 50, 64)( 51, 66)
( 52, 68)( 53, 67)( 54, 69)( 55, 71)( 56, 70)( 57, 72)( 58, 74)( 59, 73)
( 60, 75)( 61, 77)( 62, 76)( 78, 80)( 81, 83)( 84, 86)( 87, 89)( 90, 92)
( 93,108)( 94,110)( 95,109)( 96,111)( 97,113)( 98,112)( 99,114)(100,116)
(101,115)(102,117)(103,119)(104,118)(105,120)(106,122)(107,121)(123,125)
(126,128)(129,131)(132,134)(135,137)(138,153)(139,155)(140,154)(141,156)
(142,158)(143,157)(144,159)(145,161)(146,160)(147,162)(148,164)(149,163)
(150,165)(151,167)(152,166)(168,170)(171,173)(174,176)(177,179)(180,182);;
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(182)!(1,2);
s1 := Sym(182)!(  6, 15)(  7, 16)(  8, 17)(  9, 12)( 10, 13)( 11, 14)( 21, 30)
( 22, 31)( 23, 32)( 24, 27)( 25, 28)( 26, 29)( 36, 45)( 37, 46)( 38, 47)
( 39, 42)( 40, 43)( 41, 44)( 51, 60)( 52, 61)( 53, 62)( 54, 57)( 55, 58)
( 56, 59)( 66, 75)( 67, 76)( 68, 77)( 69, 72)( 70, 73)( 71, 74)( 81, 90)
( 82, 91)( 83, 92)( 84, 87)( 85, 88)( 86, 89)( 93,138)( 94,139)( 95,140)
( 96,150)( 97,151)( 98,152)( 99,147)(100,148)(101,149)(102,144)(103,145)
(104,146)(105,141)(106,142)(107,143)(108,153)(109,154)(110,155)(111,165)
(112,166)(113,167)(114,162)(115,163)(116,164)(117,159)(118,160)(119,161)
(120,156)(121,157)(122,158)(123,168)(124,169)(125,170)(126,180)(127,181)
(128,182)(129,177)(130,178)(131,179)(132,174)(133,175)(134,176)(135,171)
(136,172)(137,173);
s2 := Sym(182)!(  3, 96)(  4, 98)(  5, 97)(  6, 93)(  7, 95)(  8, 94)(  9,105)
( 10,107)( 11,106)( 12,102)( 13,104)( 14,103)( 15, 99)( 16,101)( 17,100)
( 18,128)( 19,127)( 20,126)( 21,125)( 22,124)( 23,123)( 24,137)( 25,136)
( 26,135)( 27,134)( 28,133)( 29,132)( 30,131)( 31,130)( 32,129)( 33,113)
( 34,112)( 35,111)( 36,110)( 37,109)( 38,108)( 39,122)( 40,121)( 41,120)
( 42,119)( 43,118)( 44,117)( 45,116)( 46,115)( 47,114)( 48,141)( 49,143)
( 50,142)( 51,138)( 52,140)( 53,139)( 54,150)( 55,152)( 56,151)( 57,147)
( 58,149)( 59,148)( 60,144)( 61,146)( 62,145)( 63,173)( 64,172)( 65,171)
( 66,170)( 67,169)( 68,168)( 69,182)( 70,181)( 71,180)( 72,179)( 73,178)
( 74,177)( 75,176)( 76,175)( 77,174)( 78,158)( 79,157)( 80,156)( 81,155)
( 82,154)( 83,153)( 84,167)( 85,166)( 86,165)( 87,164)( 88,163)( 89,162)
( 90,161)( 91,160)( 92,159);
s3 := Sym(182)!(  3, 18)(  4, 20)(  5, 19)(  6, 21)(  7, 23)(  8, 22)(  9, 24)
( 10, 26)( 11, 25)( 12, 27)( 13, 29)( 14, 28)( 15, 30)( 16, 32)( 17, 31)
( 33, 35)( 36, 38)( 39, 41)( 42, 44)( 45, 47)( 48, 63)( 49, 65)( 50, 64)
( 51, 66)( 52, 68)( 53, 67)( 54, 69)( 55, 71)( 56, 70)( 57, 72)( 58, 74)
( 59, 73)( 60, 75)( 61, 77)( 62, 76)( 78, 80)( 81, 83)( 84, 86)( 87, 89)
( 90, 92)( 93,108)( 94,110)( 95,109)( 96,111)( 97,113)( 98,112)( 99,114)
(100,116)(101,115)(102,117)(103,119)(104,118)(105,120)(106,122)(107,121)
(123,125)(126,128)(129,131)(132,134)(135,137)(138,153)(139,155)(140,154)
(141,156)(142,158)(143,157)(144,159)(145,161)(146,160)(147,162)(148,164)
(149,163)(150,165)(151,167)(152,166)(168,170)(171,173)(174,176)(177,179)
(180,182);
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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 >; 
 

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