Polytope of Type {20,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,18}*720b
if this polytope has a name.
Group : SmallGroup(720,394)
Rank : 3
Schlafli Type : {20,18}
Number of vertices, edges, etc : 20, 180, 18
Order of s₀s₁s₂ : 45
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   3-fold quotients : {20,6}*240b
   5-fold quotients : {4,18}*144c
   10-fold quotients : {4,9}*72
   15-fold quotients : {4,6}*48b
   30-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {20,18}*1440
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope