Polytope of Type {6,12,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,10}*1440b
if this polytope has a name.
Group : SmallGroup(1440,5282)
Rank : 4
Schlafli Type : {6,12,10}
Number of vertices, edges, etc : 6, 36, 60, 10
Order of s₀s₁s₂s₃ : 60
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,10}*720b
   3-fold quotients : {2,12,10}*480
   5-fold quotients : {6,12,2}*288b
   6-fold quotients : {2,6,10}*240
   9-fold quotients : {2,4,10}*160
   10-fold quotients : {6,6,2}*144b
   15-fold quotients : {2,12,2}*96
   18-fold quotients : {2,2,10}*80
   20-fold quotients : {6,3,2}*72
   30-fold quotients : {2,6,2}*48
   36-fold quotients : {2,2,5}*40
   45-fold quotients : {2,4,2}*32
   60-fold quotients : {2,3,2}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)
( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)
( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)( 69, 74)
( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)( 97,102)
( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)(115,120)
(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)(143,148)
(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)(171,176)
(172,177)(173,178)(174,179)(175,180);;
s1 := (  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 36)( 17, 37)( 18, 38)
( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)( 26, 41)
( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)( 49, 54)
( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)( 67, 77)
( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)
( 91,141)( 92,142)( 93,143)( 94,144)( 95,145)( 96,136)( 97,137)( 98,138)
( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)(105,150)(106,171)
(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)(113,168)(114,169)
(115,170)(116,176)(117,177)(118,178)(119,179)(120,180)(121,156)(122,157)
(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)
(131,161)(132,162)(133,163)(134,164)(135,165);;
s2 := (  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);;
s3 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)( 78, 80)
( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)( 98,100)
(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)(118,120)
(121,122)(123,125)(126,127)(128,130)(131,132)(133,135)(136,137)(138,140)
(141,142)(143,145)(146,147)(148,150)(151,152)(153,155)(156,157)(158,160)
(161,162)(163,165)(166,167)(168,170)(171,172)(173,175)(176,177)(178,180);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)!(  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)
( 23, 28)( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)
( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)
( 69, 74)( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)
( 97,102)( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)
(115,120)(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)
(143,148)(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)
(171,176)(172,177)(173,178)(174,179)(175,180);
s1 := Sym(180)!(  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 36)( 17, 37)
( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)
( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)
( 49, 54)( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)
( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)
( 75, 90)( 91,141)( 92,142)( 93,143)( 94,144)( 95,145)( 96,136)( 97,137)
( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,171)(107,172)(108,173)(109,174)(110,175)(111,166)(112,167)(113,168)
(114,169)(115,170)(116,176)(117,177)(118,178)(119,179)(120,180)(121,156)
(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)
(130,155)(131,161)(132,162)(133,163)(134,164)(135,165);
s2 := 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);
s3 := Sym(180)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)
( 78, 80)( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91, 92)( 93, 95)( 96, 97)
( 98,100)(101,102)(103,105)(106,107)(108,110)(111,112)(113,115)(116,117)
(118,120)(121,122)(123,125)(126,127)(128,130)(131,132)(133,135)(136,137)
(138,140)(141,142)(143,145)(146,147)(148,150)(151,152)(153,155)(156,157)
(158,160)(161,162)(163,165)(166,167)(168,170)(171,172)(173,175)(176,177)
(178,180);
poly := sub<Sym(180)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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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