Polytope of Type {6,60}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,60}*720c
if this polytope has a name.
Group : SmallGroup(720,673)
Rank : 3
Schlafli Type : {6,60}
Number of vertices, edges, etc : 6, 180, 60
Order of s₀s₁s₂ : 60
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,60,2} of size 1440
Vertex Figure Of :
   {2,6,60} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,30}*360c
   3-fold quotients : {2,60}*240
   4-fold quotients : {6,15}*180
   5-fold quotients : {6,12}*144b
   6-fold quotients : {2,30}*120
   9-fold quotients : {2,20}*80
   10-fold quotients : {6,6}*72b
   12-fold quotients : {2,15}*60
   15-fold quotients : {2,12}*48
   18-fold quotients : {2,10}*40
   20-fold quotients : {6,3}*36
   30-fold quotients : {2,6}*24
   36-fold quotients : {2,5}*20
   45-fold quotients : {2,4}*16
   60-fold quotients : {2,3}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,120}*1440c, {12,60}*1440c
Permutation Representation (GAP) :

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