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Polytope of Type {12,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,30}*720b
Also Known As : {12,30|2}. if this polytope has another name.
Group : SmallGroup(720,672)
Rank : 3
Schlafli Type : {12,30}
Number of vertices, edges, etc : 12, 180, 30
Order of s0s1s2 : 60
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,30,2} of size 1440
Vertex Figure Of :
   {2,12,30} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,30}*360b
   3-fold quotients : {12,10}*240, {4,30}*240a
   5-fold quotients : {12,6}*144a
   6-fold quotients : {6,10}*120, {2,30}*120
   9-fold quotients : {4,10}*80
   10-fold quotients : {6,6}*72a
   12-fold quotients : {2,15}*60
   15-fold quotients : {12,2}*48, {4,6}*48a
   18-fold quotients : {2,10}*40
   30-fold quotients : {2,6}*24, {6,2}*24
   36-fold quotients : {2,5}*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 : {24,30}*1440b, {12,60}*1440b
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)( 91,136)( 92,137)
( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)(100,145)
(101,146)(102,147)(103,148)(104,149)(105,150)(106,166)(107,167)(108,168)
(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)
(117,177)(118,178)(119,179)(120,180)(121,151)(122,152)(123,153)(124,154)
(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)
(133,163)(134,164)(135,165);;
s1 := (  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,116)(  7,120)(  8,119)
(  9,118)( 10,117)( 11,111)( 12,115)( 13,114)( 14,113)( 15,112)( 16, 91)
( 17, 95)( 18, 94)( 19, 93)( 20, 92)( 21,101)( 22,105)( 23,104)( 24,103)
( 25,102)( 26, 96)( 27,100)( 28, 99)( 29, 98)( 30, 97)( 31,121)( 32,125)
( 33,124)( 34,123)( 35,122)( 36,131)( 37,135)( 38,134)( 39,133)( 40,132)
( 41,126)( 42,130)( 43,129)( 44,128)( 45,127)( 46,151)( 47,155)( 48,154)
( 49,153)( 50,152)( 51,161)( 52,165)( 53,164)( 54,163)( 55,162)( 56,156)
( 57,160)( 58,159)( 59,158)( 60,157)( 61,136)( 62,140)( 63,139)( 64,138)
( 65,137)( 66,146)( 67,150)( 68,149)( 69,148)( 70,147)( 71,141)( 72,145)
( 73,144)( 74,143)( 75,142)( 76,166)( 77,170)( 78,169)( 79,168)( 80,167)
( 81,176)( 82,180)( 83,179)( 84,178)( 85,177)( 86,171)( 87,175)( 88,174)
( 89,173)( 90,172);;
s2 := (  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 22)
( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 27)( 28, 30)( 31, 37)( 32, 36)
( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 52)( 47, 51)( 48, 55)
( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 67)( 62, 66)( 63, 70)( 64, 69)
( 65, 68)( 71, 72)( 73, 75)( 76, 82)( 77, 81)( 78, 85)( 79, 84)( 80, 83)
( 86, 87)( 88, 90)( 91, 97)( 92, 96)( 93,100)( 94, 99)( 95, 98)(101,102)
(103,105)(106,112)(107,111)(108,115)(109,114)(110,113)(116,117)(118,120)
(121,127)(122,126)(123,130)(124,129)(125,128)(131,132)(133,135)(136,142)
(137,141)(138,145)(139,144)(140,143)(146,147)(148,150)(151,157)(152,156)
(153,160)(154,159)(155,158)(161,162)(163,165)(166,172)(167,171)(168,175)
(169,174)(170,173)(176,177)(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, 
s0*s1*s0*s1*s0*s1*s0*s1*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 ];;
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)( 91,136)
( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)
(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,166)(107,167)
(108,168)(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)
(116,176)(117,177)(118,178)(119,179)(120,180)(121,151)(122,152)(123,153)
(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)
(132,162)(133,163)(134,164)(135,165);
s1 := Sym(180)!(  1,106)(  2,110)(  3,109)(  4,108)(  5,107)(  6,116)(  7,120)
(  8,119)(  9,118)( 10,117)( 11,111)( 12,115)( 13,114)( 14,113)( 15,112)
( 16, 91)( 17, 95)( 18, 94)( 19, 93)( 20, 92)( 21,101)( 22,105)( 23,104)
( 24,103)( 25,102)( 26, 96)( 27,100)( 28, 99)( 29, 98)( 30, 97)( 31,121)
( 32,125)( 33,124)( 34,123)( 35,122)( 36,131)( 37,135)( 38,134)( 39,133)
( 40,132)( 41,126)( 42,130)( 43,129)( 44,128)( 45,127)( 46,151)( 47,155)
( 48,154)( 49,153)( 50,152)( 51,161)( 52,165)( 53,164)( 54,163)( 55,162)
( 56,156)( 57,160)( 58,159)( 59,158)( 60,157)( 61,136)( 62,140)( 63,139)
( 64,138)( 65,137)( 66,146)( 67,150)( 68,149)( 69,148)( 70,147)( 71,141)
( 72,145)( 73,144)( 74,143)( 75,142)( 76,166)( 77,170)( 78,169)( 79,168)
( 80,167)( 81,176)( 82,180)( 83,179)( 84,178)( 85,177)( 86,171)( 87,175)
( 88,174)( 89,173)( 90,172);
s2 := Sym(180)!(  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)
( 16, 22)( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 27)( 28, 30)( 31, 37)
( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 52)( 47, 51)
( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 67)( 62, 66)( 63, 70)
( 64, 69)( 65, 68)( 71, 72)( 73, 75)( 76, 82)( 77, 81)( 78, 85)( 79, 84)
( 80, 83)( 86, 87)( 88, 90)( 91, 97)( 92, 96)( 93,100)( 94, 99)( 95, 98)
(101,102)(103,105)(106,112)(107,111)(108,115)(109,114)(110,113)(116,117)
(118,120)(121,127)(122,126)(123,130)(124,129)(125,128)(131,132)(133,135)
(136,142)(137,141)(138,145)(139,144)(140,143)(146,147)(148,150)(151,157)
(152,156)(153,160)(154,159)(155,158)(161,162)(163,165)(166,172)(167,171)
(168,175)(169,174)(170,173)(176,177)(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, 
s0*s1*s0*s1*s0*s1*s0*s1*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 >; 
 
References : None.
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