Polytope of Type {18,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,20}*720b
if this polytope has a name.
Group : SmallGroup(720,394)
Rank : 3
Schlafli Type : {18,20}
Number of vertices, edges, etc : 18, 180, 20
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 :
   {18,20,2} of size 1440
Vertex Figure Of :
   {2,18,20} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,20}*240b
   5-fold quotients : {18,4}*144c
   10-fold quotients : {9,4}*72
   15-fold quotients : {6,4}*48b
   30-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,20}*1440
Irregular Quotients (of which this is a minimal cover):
   None.

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