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Polytope of Type {30,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,12}*720c
if this polytope has a name.
Group : SmallGroup(720,687)
Rank : 3
Schlafli Type : {30,12}
Number of vertices, edges, etc : 30, 180, 12
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 :
   {30,12,2} of size 1440
Vertex Figure Of :
   {2,30,12} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {30,6}*360c
   3-fold quotients : {30,4}*240a
   4-fold quotients : {15,6}*180
   5-fold quotients : {6,12}*144c
   6-fold quotients : {30,2}*120
   9-fold quotients : {10,4}*80
   10-fold quotients : {6,6}*72c
   12-fold quotients : {15,2}*60
   15-fold quotients : {6,4}*48a
   18-fold quotients : {10,2}*40
   20-fold quotients : {3,6}*36
   30-fold quotients : {6,2}*24
   36-fold quotients : {5,2}*20
   45-fold quotients : {2,4}*16
   60-fold quotients : {3,2}*12
   90-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {60,12}*1440c, {30,24}*1440c
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  6, 11)(  7, 15)(  8, 14)(  9, 13)( 10, 12)( 16, 31)
( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 41)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 36)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 47, 50)( 48, 49)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 76)( 62, 80)( 63, 79)
( 64, 78)( 65, 77)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 81)
( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 92, 95)( 93, 94)( 96,101)( 97,105)
( 98,104)( 99,103)(100,102)(106,121)(107,125)(108,124)(109,123)(110,122)
(111,131)(112,135)(113,134)(114,133)(115,132)(116,126)(117,130)(118,129)
(119,128)(120,127)(137,140)(138,139)(141,146)(142,150)(143,149)(144,148)
(145,147)(151,166)(152,170)(153,169)(154,168)(155,167)(156,176)(157,180)
(158,179)(159,178)(160,177)(161,171)(162,175)(163,174)(164,173)(165,172);;
s1 := (  1,112)(  2,111)(  3,115)(  4,114)(  5,113)(  6,107)(  7,106)(  8,110)
(  9,109)( 10,108)( 11,117)( 12,116)( 13,120)( 14,119)( 15,118)( 16, 97)
( 17, 96)( 18,100)( 19, 99)( 20, 98)( 21, 92)( 22, 91)( 23, 95)( 24, 94)
( 25, 93)( 26,102)( 27,101)( 28,105)( 29,104)( 30,103)( 31,127)( 32,126)
( 33,130)( 34,129)( 35,128)( 36,122)( 37,121)( 38,125)( 39,124)( 40,123)
( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,157)( 47,156)( 48,160)
( 49,159)( 50,158)( 51,152)( 52,151)( 53,155)( 54,154)( 55,153)( 56,162)
( 57,161)( 58,165)( 59,164)( 60,163)( 61,142)( 62,141)( 63,145)( 64,144)
( 65,143)( 66,137)( 67,136)( 68,140)( 69,139)( 70,138)( 71,147)( 72,146)
( 73,150)( 74,149)( 75,148)( 76,172)( 77,171)( 78,175)( 79,174)( 80,173)
( 81,167)( 82,166)( 83,170)( 84,169)( 85,168)( 86,177)( 87,176)( 88,180)
( 89,179)( 90,178);;
s2 := ( 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);;
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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)
( 16, 31)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 41)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 36)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 47, 50)
( 48, 49)( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 76)( 62, 80)
( 63, 79)( 64, 78)( 65, 77)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)
( 71, 81)( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 92, 95)( 93, 94)( 96,101)
( 97,105)( 98,104)( 99,103)(100,102)(106,121)(107,125)(108,124)(109,123)
(110,122)(111,131)(112,135)(113,134)(114,133)(115,132)(116,126)(117,130)
(118,129)(119,128)(120,127)(137,140)(138,139)(141,146)(142,150)(143,149)
(144,148)(145,147)(151,166)(152,170)(153,169)(154,168)(155,167)(156,176)
(157,180)(158,179)(159,178)(160,177)(161,171)(162,175)(163,174)(164,173)
(165,172);
s1 := Sym(180)!(  1,112)(  2,111)(  3,115)(  4,114)(  5,113)(  6,107)(  7,106)
(  8,110)(  9,109)( 10,108)( 11,117)( 12,116)( 13,120)( 14,119)( 15,118)
( 16, 97)( 17, 96)( 18,100)( 19, 99)( 20, 98)( 21, 92)( 22, 91)( 23, 95)
( 24, 94)( 25, 93)( 26,102)( 27,101)( 28,105)( 29,104)( 30,103)( 31,127)
( 32,126)( 33,130)( 34,129)( 35,128)( 36,122)( 37,121)( 38,125)( 39,124)
( 40,123)( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,157)( 47,156)
( 48,160)( 49,159)( 50,158)( 51,152)( 52,151)( 53,155)( 54,154)( 55,153)
( 56,162)( 57,161)( 58,165)( 59,164)( 60,163)( 61,142)( 62,141)( 63,145)
( 64,144)( 65,143)( 66,137)( 67,136)( 68,140)( 69,139)( 70,138)( 71,147)
( 72,146)( 73,150)( 74,149)( 75,148)( 76,172)( 77,171)( 78,175)( 79,174)
( 80,173)( 81,167)( 82,166)( 83,170)( 84,169)( 85,168)( 86,177)( 87,176)
( 88,180)( 89,179)( 90,178);
s2 := 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);
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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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