Polytope of Type {12,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,10}*1440c
if this polytope has a name.
Group : SmallGroup(1440,5358)
Rank : 4
Schlafli Type : {12,6,10}
Number of vertices, edges, etc : 12, 36, 30, 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 : {4,6,10}*480a
   5-fold quotients : {12,6,2}*288c
   6-fold quotients : {2,6,10}*240
   9-fold quotients : {4,2,10}*160
   10-fold quotients : {6,6,2}*144b
   15-fold quotients : {4,6,2}*96a
   18-fold quotients : {4,2,5}*80, {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 : {4,2,2}*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)( 91,136)( 92,137)
( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)( 98,148)( 99,149)(100,150)
(101,141)(102,142)(103,143)(104,144)(105,145)(106,151)(107,152)(108,153)
(109,154)(110,155)(111,161)(112,162)(113,163)(114,164)(115,165)(116,156)
(117,157)(118,158)(119,159)(120,160)(121,166)(122,167)(123,168)(124,169)
(125,170)(126,176)(127,177)(128,178)(129,179)(130,180)(131,171)(132,172)
(133,173)(134,174)(135,175);;
s1 := (  1, 96)(  2, 97)(  3, 98)(  4, 99)(  5,100)(  6, 91)(  7, 92)(  8, 93)
(  9, 94)( 10, 95)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,126)
( 17,127)( 18,128)( 19,129)( 20,130)( 21,121)( 22,122)( 23,123)( 24,124)
( 25,125)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)
( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46,141)( 47,142)( 48,143)
( 49,144)( 50,145)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)( 64,174)
( 65,175)( 66,166)( 67,167)( 68,168)( 69,169)( 70,170)( 71,176)( 72,177)
( 73,178)( 74,179)( 75,180)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)
( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,161)( 87,162)( 88,163)
( 89,164)( 90,165);;
s2 := (  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)( 91,106)( 92,110)( 93,109)( 94,108)
( 95,107)( 96,116)( 97,120)( 98,119)( 99,118)(100,117)(101,111)(102,115)
(103,114)(104,113)(105,112)(122,125)(123,124)(126,131)(127,135)(128,134)
(129,133)(130,132)(136,151)(137,155)(138,154)(139,153)(140,152)(141,161)
(142,165)(143,164)(144,163)(145,162)(146,156)(147,160)(148,159)(149,158)
(150,157)(167,170)(168,169)(171,176)(172,180)(173,179)(174,178)(175,177);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)( 91,136)
( 92,137)( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)( 98,148)( 99,149)
(100,150)(101,141)(102,142)(103,143)(104,144)(105,145)(106,151)(107,152)
(108,153)(109,154)(110,155)(111,161)(112,162)(113,163)(114,164)(115,165)
(116,156)(117,157)(118,158)(119,159)(120,160)(121,166)(122,167)(123,168)
(124,169)(125,170)(126,176)(127,177)(128,178)(129,179)(130,180)(131,171)
(132,172)(133,173)(134,174)(135,175);
s1 := Sym(180)!(  1, 96)(  2, 97)(  3, 98)(  4, 99)(  5,100)(  6, 91)(  7, 92)
(  8, 93)(  9, 94)( 10, 95)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)
( 16,126)( 17,127)( 18,128)( 19,129)( 20,130)( 21,121)( 22,122)( 23,123)
( 24,124)( 25,125)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,111)
( 32,112)( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)
( 40,110)( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46,141)( 47,142)
( 48,143)( 49,144)( 50,145)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)
( 64,174)( 65,175)( 66,166)( 67,167)( 68,168)( 69,169)( 70,170)( 71,176)
( 72,177)( 73,178)( 74,179)( 75,180)( 76,156)( 77,157)( 78,158)( 79,159)
( 80,160)( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,161)( 87,162)
( 88,163)( 89,164)( 90,165);
s2 := 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)( 91,106)( 92,110)( 93,109)
( 94,108)( 95,107)( 96,116)( 97,120)( 98,119)( 99,118)(100,117)(101,111)
(102,115)(103,114)(104,113)(105,112)(122,125)(123,124)(126,131)(127,135)
(128,134)(129,133)(130,132)(136,151)(137,155)(138,154)(139,153)(140,152)
(141,161)(142,165)(143,164)(144,163)(145,162)(146,156)(147,160)(148,159)
(149,158)(150,157)(167,170)(168,169)(171,176)(172,180)(173,179)(174,178)
(175,177);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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