Polytope of Type {6,12,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,10}*1440c
if this polytope has a name.
Group : SmallGroup(1440,5358)
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}*720c
3-fold quotients : {6,4,10}*480
4-fold quotients : {3,6,10}*360
5-fold quotients : {6,12,2}*288c
6-fold quotients : {6,2,10}*240
9-fold quotients : {2,4,10}*160
10-fold quotients : {6,6,2}*144c
12-fold quotients : {3,2,10}*120, {6,2,5}*120
15-fold quotients : {6,4,2}*96a
18-fold quotients : {2,2,10}*80
20-fold quotients : {3,6,2}*72
24-fold quotients : {3,2,5}*60
30-fold quotients : {6,2,2}*48
36-fold quotients : {2,2,5}*40
45-fold quotients : {2,4,2}*32
60-fold quotients : {3,2,2}*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)( 16, 31)( 17, 32)( 18, 33)
( 19, 34)( 20, 35)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)( 26, 36)
( 27, 37)( 28, 38)( 29, 39)( 30, 40)( 51, 56)( 52, 57)( 53, 58)( 54, 59)
( 55, 60)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 86)( 67, 87)
( 68, 88)( 69, 89)( 70, 90)( 71, 81)( 72, 82)( 73, 83)( 74, 84)( 75, 85)
( 96,101)( 97,102)( 98,103)( 99,104)(100,105)(106,121)(107,122)(108,123)
(109,124)(110,125)(111,131)(112,132)(113,133)(114,134)(115,135)(116,126)
(117,127)(118,128)(119,129)(120,130)(141,146)(142,147)(143,148)(144,149)
(145,150)(151,166)(152,167)(153,168)(154,169)(155,170)(156,176)(157,177)
(158,178)(159,179)(160,180)(161,171)(162,172)(163,173)(164,174)(165,175);;
s1 := ( 1,111)( 2,112)( 3,113)( 4,114)( 5,115)( 6,106)( 7,107)( 8,108)
( 9,109)( 10,110)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21, 91)( 22, 92)( 23, 93)( 24, 94)
( 25, 95)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,126)( 32,127)
( 33,128)( 34,129)( 35,130)( 36,121)( 37,122)( 38,123)( 39,124)( 40,125)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,156)( 47,157)( 48,158)
( 49,159)( 50,160)( 51,151)( 52,152)( 53,153)( 54,154)( 55,155)( 56,161)
( 57,162)( 58,163)( 59,164)( 60,165)( 61,141)( 62,142)( 63,143)( 64,144)
( 65,145)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,146)( 72,147)
( 73,148)( 74,149)( 75,150)( 76,171)( 77,172)( 78,173)( 79,174)( 80,175)
( 81,166)( 82,167)( 83,168)( 84,169)( 85,170)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180);;
s2 := ( 2, 5)( 3, 4)( 6, 11)( 7, 15)( 8, 14)( 9, 13)( 10, 12)( 17, 20)
( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)( 33, 34)
( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)( 51, 56)
( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)( 67, 75)
( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)( 83, 89)
( 84, 88)( 85, 87)( 91,136)( 92,140)( 93,139)( 94,138)( 95,137)( 96,146)
( 97,150)( 98,149)( 99,148)(100,147)(101,141)(102,145)(103,144)(104,143)
(105,142)(106,151)(107,155)(108,154)(109,153)(110,152)(111,161)(112,165)
(113,164)(114,163)(115,162)(116,156)(117,160)(118,159)(119,158)(120,157)
(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);;
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*s2*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(180)!( 6, 11)( 7, 12)( 8, 13)( 9, 14)( 10, 15)( 16, 31)( 17, 32)
( 18, 33)( 19, 34)( 20, 35)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)
( 26, 36)( 27, 37)( 28, 38)( 29, 39)( 30, 40)( 51, 56)( 52, 57)( 53, 58)
( 54, 59)( 55, 60)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 86)
( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71, 81)( 72, 82)( 73, 83)( 74, 84)
( 75, 85)( 96,101)( 97,102)( 98,103)( 99,104)(100,105)(106,121)(107,122)
(108,123)(109,124)(110,125)(111,131)(112,132)(113,133)(114,134)(115,135)
(116,126)(117,127)(118,128)(119,129)(120,130)(141,146)(142,147)(143,148)
(144,149)(145,150)(151,166)(152,167)(153,168)(154,169)(155,170)(156,176)
(157,177)(158,178)(159,179)(160,180)(161,171)(162,172)(163,173)(164,174)
(165,175);
s1 := Sym(180)!( 1,111)( 2,112)( 3,113)( 4,114)( 5,115)( 6,106)( 7,107)
( 8,108)( 9,109)( 10,110)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21, 91)( 22, 92)( 23, 93)
( 24, 94)( 25, 95)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,126)
( 32,127)( 33,128)( 34,129)( 35,130)( 36,121)( 37,122)( 38,123)( 39,124)
( 40,125)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,156)( 47,157)
( 48,158)( 49,159)( 50,160)( 51,151)( 52,152)( 53,153)( 54,154)( 55,155)
( 56,161)( 57,162)( 58,163)( 59,164)( 60,165)( 61,141)( 62,142)( 63,143)
( 64,144)( 65,145)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,146)
( 72,147)( 73,148)( 74,149)( 75,150)( 76,171)( 77,172)( 78,173)( 79,174)
( 80,175)( 81,166)( 82,167)( 83,168)( 84,169)( 85,170)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180);
s2 := Sym(180)!( 2, 5)( 3, 4)( 6, 11)( 7, 15)( 8, 14)( 9, 13)( 10, 12)
( 17, 20)( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)
( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)
( 83, 89)( 84, 88)( 85, 87)( 91,136)( 92,140)( 93,139)( 94,138)( 95,137)
( 96,146)( 97,150)( 98,149)( 99,148)(100,147)(101,141)(102,145)(103,144)
(104,143)(105,142)(106,151)(107,155)(108,154)(109,153)(110,152)(111,161)
(112,165)(113,164)(114,163)(115,162)(116,156)(117,160)(118,159)(119,158)
(120,157)(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);
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*s2*s1*s2*s0*s1*s2*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 >;
References : None.
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