Polytope of Type {2,20,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,20,18}*1440b
if this polytope has a name.
Group : SmallGroup(1440,4569)
Rank : 4
Schlafli Type : {2,20,18}
Number of vertices, edges, etc : 2, 20, 180, 18
Order of s₀s₁s₂s₃ : 90
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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) :
   3-fold quotients : {2,20,6}*480b
   5-fold quotients : {2,4,18}*288c
   10-fold quotients : {2,4,9}*144
   15-fold quotients : {2,4,6}*96b
   30-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

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