Polytope of Type {20,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,18}*720a
Also Known As : {20,18|2}. if this polytope has another name.
Group : SmallGroup(720,137)
Rank : 3
Schlafli Type : {20,18}
Number of vertices, edges, etc : 20, 180, 18
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 :
   {20,18,2} of size 1440
Vertex Figure Of :
   {2,20,18} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,18}*360
   3-fold quotients : {20,6}*240a
   5-fold quotients : {4,18}*144a
   6-fold quotients : {10,6}*120
   9-fold quotients : {20,2}*80
   10-fold quotients : {2,18}*72
   15-fold quotients : {4,6}*48a
   18-fold quotients : {10,2}*40
   20-fold quotients : {2,9}*36
   30-fold quotients : {2,6}*24
   36-fold quotients : {5,2}*20
   45-fold quotients : {4,2}*16
   60-fold quotients : {2,3}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {40,18}*1440, {20,36}*1440
Permutation Representation (GAP) :

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