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Polytope of Type {60,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {60,6}*720b
Also Known As : {60,6|2}. if this polytope has another name.
Group : SmallGroup(720,673)
Rank : 3
Schlafli Type : {60,6}
Number of vertices, edges, etc : 60, 180, 6
Order of s0s1s2 : 60
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {60,6,2} of size 1440
Vertex Figure Of :
   {2,60,6} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {30,6}*360b
   3-fold quotients : {20,6}*240a, {60,2}*240
   5-fold quotients : {12,6}*144a
   6-fold quotients : {10,6}*120, {30,2}*120
   9-fold quotients : {20,2}*80
   10-fold quotients : {6,6}*72a
   12-fold quotients : {15,2}*60
   15-fold quotients : {12,2}*48, {4,6}*48a
   18-fold quotients : {10,2}*40
   30-fold quotients : {2,6}*24, {6,2}*24
   36-fold quotients : {5,2}*20
   45-fold quotients : {4,2}*16
   60-fold quotients : {2,3}*12, {3,2}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {120,6}*1440b, {60,12}*1440b
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)( 17, 20)
( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)( 33, 34)
( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)( 51, 56)
( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)( 67, 75)
( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)( 83, 89)
( 84, 88)( 85, 87)( 91,136)( 92,140)( 93,139)( 94,138)( 95,137)( 96,146)
( 97,150)( 98,149)( 99,148)(100,147)(101,141)(102,145)(103,144)(104,143)
(105,142)(106,151)(107,155)(108,154)(109,153)(110,152)(111,161)(112,165)
(113,164)(114,163)(115,162)(116,156)(117,160)(118,159)(119,158)(120,157)
(121,166)(122,170)(123,169)(124,168)(125,167)(126,176)(127,180)(128,179)
(129,178)(130,177)(131,171)(132,175)(133,174)(134,173)(135,172);;
s1 := (  1, 97)(  2, 96)(  3,100)(  4, 99)(  5, 98)(  6, 92)(  7, 91)(  8, 95)
(  9, 94)( 10, 93)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)( 16,127)
( 17,126)( 18,130)( 19,129)( 20,128)( 21,122)( 22,121)( 23,125)( 24,124)
( 25,123)( 26,132)( 27,131)( 28,135)( 29,134)( 30,133)( 31,112)( 32,111)
( 33,115)( 34,114)( 35,113)( 36,107)( 37,106)( 38,110)( 39,109)( 40,108)
( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46,142)( 47,141)( 48,145)
( 49,144)( 50,143)( 51,137)( 52,136)( 53,140)( 54,139)( 55,138)( 56,147)
( 57,146)( 58,150)( 59,149)( 60,148)( 61,172)( 62,171)( 63,175)( 64,174)
( 65,173)( 66,167)( 67,166)( 68,170)( 69,169)( 70,168)( 71,177)( 72,176)
( 73,180)( 74,179)( 75,178)( 76,157)( 77,156)( 78,160)( 79,159)( 80,158)
( 81,152)( 82,151)( 83,155)( 84,154)( 85,153)( 86,162)( 87,161)( 88,165)
( 89,164)( 90,163);;
s2 := (  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)(  8, 23)
(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 46, 61)
( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 51, 66)( 52, 67)( 53, 68)( 54, 69)
( 55, 70)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 91,106)( 92,107)
( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)
(101,116)(102,117)(103,118)(104,119)(105,120)(136,151)(137,152)(138,153)
(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)
(147,162)(148,163)(149,164)(150,165);;
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, 
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*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*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)!(  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)
( 17, 20)( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)
( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)
( 83, 89)( 84, 88)( 85, 87)( 91,136)( 92,140)( 93,139)( 94,138)( 95,137)
( 96,146)( 97,150)( 98,149)( 99,148)(100,147)(101,141)(102,145)(103,144)
(104,143)(105,142)(106,151)(107,155)(108,154)(109,153)(110,152)(111,161)
(112,165)(113,164)(114,163)(115,162)(116,156)(117,160)(118,159)(119,158)
(120,157)(121,166)(122,170)(123,169)(124,168)(125,167)(126,176)(127,180)
(128,179)(129,178)(130,177)(131,171)(132,175)(133,174)(134,173)(135,172);
s1 := Sym(180)!(  1, 97)(  2, 96)(  3,100)(  4, 99)(  5, 98)(  6, 92)(  7, 91)
(  8, 95)(  9, 94)( 10, 93)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)
( 16,127)( 17,126)( 18,130)( 19,129)( 20,128)( 21,122)( 22,121)( 23,125)
( 24,124)( 25,123)( 26,132)( 27,131)( 28,135)( 29,134)( 30,133)( 31,112)
( 32,111)( 33,115)( 34,114)( 35,113)( 36,107)( 37,106)( 38,110)( 39,109)
( 40,108)( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46,142)( 47,141)
( 48,145)( 49,144)( 50,143)( 51,137)( 52,136)( 53,140)( 54,139)( 55,138)
( 56,147)( 57,146)( 58,150)( 59,149)( 60,148)( 61,172)( 62,171)( 63,175)
( 64,174)( 65,173)( 66,167)( 67,166)( 68,170)( 69,169)( 70,168)( 71,177)
( 72,176)( 73,180)( 74,179)( 75,178)( 76,157)( 77,156)( 78,160)( 79,159)
( 80,158)( 81,152)( 82,151)( 83,155)( 84,154)( 85,153)( 86,162)( 87,161)
( 88,165)( 89,164)( 90,163);
s2 := Sym(180)!(  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)
(  8, 23)(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)
( 46, 61)( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 51, 66)( 52, 67)( 53, 68)
( 54, 69)( 55, 70)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 91,106)
( 92,107)( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)
(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(136,151)(137,152)
(138,153)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)
(146,161)(147,162)(148,163)(149,164)(150,165);
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, 
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*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*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.
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