Polytope of Type {18,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,20}*720a
Also Known As : {18,20|2}. if this polytope has another name.
Group : SmallGroup(720,137)
Rank : 3
Schlafli Type : {18,20}
Number of vertices, edges, etc : 18, 180, 20
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 :
   {18,20,2} of size 1440
Vertex Figure Of :
   {2,18,20} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,10}*360
   3-fold quotients : {6,20}*240a
   5-fold quotients : {18,4}*144a
   6-fold quotients : {6,10}*120
   9-fold quotients : {2,20}*80
   10-fold quotients : {18,2}*72
   15-fold quotients : {6,4}*48a
   18-fold quotients : {2,10}*40
   20-fold quotients : {9,2}*36
   30-fold quotients : {6,2}*24
   36-fold quotients : {2,5}*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 : {18,40}*1440, {36,20}*1440
Permutation Representation (GAP) :

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