Polytope of Type {12,10,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,10,6}*1440
Also Known As : {{12,10|2},{10,6|2}}. if this polytope has another name.
Group : SmallGroup(1440,5282)
Rank : 4
Schlafli Type : {12,10,6}
Number of vertices, edges, etc : 12, 60, 30, 6
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 : {6,10,6}*720
3-fold quotients : {12,10,2}*480, {4,10,6}*480
5-fold quotients : {12,2,6}*288
6-fold quotients : {2,10,6}*240, {6,10,2}*240
9-fold quotients : {4,10,2}*160
10-fold quotients : {12,2,3}*144, {6,2,6}*144
15-fold quotients : {12,2,2}*96, {4,2,6}*96
18-fold quotients : {2,10,2}*80
20-fold quotients : {3,2,6}*72, {6,2,3}*72
30-fold quotients : {4,2,3}*48, {2,2,6}*48, {6,2,2}*48
36-fold quotients : {2,5,2}*40
40-fold quotients : {3,2,3}*36
45-fold quotients : {4,2,2}*32
60-fold quotients : {2,2,3}*24, {3,2,2}*24
90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
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)( 6,111)( 7,115)( 8,114)
( 9,113)( 10,112)( 11,116)( 12,120)( 13,119)( 14,118)( 15,117)( 16, 91)
( 17, 95)( 18, 94)( 19, 93)( 20, 92)( 21, 96)( 22,100)( 23, 99)( 24, 98)
( 25, 97)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,121)( 32,125)
( 33,124)( 34,123)( 35,122)( 36,126)( 37,130)( 38,129)( 39,128)( 40,127)
( 41,131)( 42,135)( 43,134)( 44,133)( 45,132)( 46,151)( 47,155)( 48,154)
( 49,153)( 50,152)( 51,156)( 52,160)( 53,159)( 54,158)( 55,157)( 56,161)
( 57,165)( 58,164)( 59,163)( 60,162)( 61,136)( 62,140)( 63,139)( 64,138)
( 65,137)( 66,141)( 67,145)( 68,144)( 69,143)( 70,142)( 71,146)( 72,150)
( 73,149)( 74,148)( 75,147)( 76,166)( 77,170)( 78,169)( 79,168)( 80,167)
( 81,171)( 82,175)( 83,174)( 84,173)( 85,172)( 86,176)( 87,180)( 88,179)
( 89,178)( 90,177);;
s2 := ( 1, 2)( 3, 5)( 6, 12)( 7, 11)( 8, 15)( 9, 14)( 10, 13)( 16, 17)
( 18, 20)( 21, 27)( 22, 26)( 23, 30)( 24, 29)( 25, 28)( 31, 32)( 33, 35)
( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 47)( 48, 50)( 51, 57)
( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 62)( 63, 65)( 66, 72)( 67, 71)
( 68, 75)( 69, 74)( 70, 73)( 76, 77)( 78, 80)( 81, 87)( 82, 86)( 83, 90)
( 84, 89)( 85, 88)( 91, 92)( 93, 95)( 96,102)( 97,101)( 98,105)( 99,104)
(100,103)(106,107)(108,110)(111,117)(112,116)(113,120)(114,119)(115,118)
(121,122)(123,125)(126,132)(127,131)(128,135)(129,134)(130,133)(136,137)
(138,140)(141,147)(142,146)(143,150)(144,149)(145,148)(151,152)(153,155)
(156,162)(157,161)(158,165)(159,164)(160,163)(166,167)(168,170)(171,177)
(172,176)(173,180)(174,179)(175,178);;
s3 := ( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 21)( 17, 22)( 18, 23)
( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 46, 51)
( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 66)( 62, 67)( 63, 68)( 64, 69)
( 65, 70)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 91, 96)( 92, 97)
( 93, 98)( 94, 99)( 95,100)(106,111)(107,112)(108,113)(109,114)(110,115)
(121,126)(122,127)(123,128)(124,129)(125,130)(136,141)(137,142)(138,143)
(139,144)(140,145)(151,156)(152,157)(153,158)(154,159)(155,160)(166,171)
(167,172)(168,173)(169,174)(170,175);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, 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,
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)!( 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)( 6,111)( 7,115)
( 8,114)( 9,113)( 10,112)( 11,116)( 12,120)( 13,119)( 14,118)( 15,117)
( 16, 91)( 17, 95)( 18, 94)( 19, 93)( 20, 92)( 21, 96)( 22,100)( 23, 99)
( 24, 98)( 25, 97)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,121)
( 32,125)( 33,124)( 34,123)( 35,122)( 36,126)( 37,130)( 38,129)( 39,128)
( 40,127)( 41,131)( 42,135)( 43,134)( 44,133)( 45,132)( 46,151)( 47,155)
( 48,154)( 49,153)( 50,152)( 51,156)( 52,160)( 53,159)( 54,158)( 55,157)
( 56,161)( 57,165)( 58,164)( 59,163)( 60,162)( 61,136)( 62,140)( 63,139)
( 64,138)( 65,137)( 66,141)( 67,145)( 68,144)( 69,143)( 70,142)( 71,146)
( 72,150)( 73,149)( 74,148)( 75,147)( 76,166)( 77,170)( 78,169)( 79,168)
( 80,167)( 81,171)( 82,175)( 83,174)( 84,173)( 85,172)( 86,176)( 87,180)
( 88,179)( 89,178)( 90,177);
s2 := Sym(180)!( 1, 2)( 3, 5)( 6, 12)( 7, 11)( 8, 15)( 9, 14)( 10, 13)
( 16, 17)( 18, 20)( 21, 27)( 22, 26)( 23, 30)( 24, 29)( 25, 28)( 31, 32)
( 33, 35)( 36, 42)( 37, 41)( 38, 45)( 39, 44)( 40, 43)( 46, 47)( 48, 50)
( 51, 57)( 52, 56)( 53, 60)( 54, 59)( 55, 58)( 61, 62)( 63, 65)( 66, 72)
( 67, 71)( 68, 75)( 69, 74)( 70, 73)( 76, 77)( 78, 80)( 81, 87)( 82, 86)
( 83, 90)( 84, 89)( 85, 88)( 91, 92)( 93, 95)( 96,102)( 97,101)( 98,105)
( 99,104)(100,103)(106,107)(108,110)(111,117)(112,116)(113,120)(114,119)
(115,118)(121,122)(123,125)(126,132)(127,131)(128,135)(129,134)(130,133)
(136,137)(138,140)(141,147)(142,146)(143,150)(144,149)(145,148)(151,152)
(153,155)(156,162)(157,161)(158,165)(159,164)(160,163)(166,167)(168,170)
(171,177)(172,176)(173,180)(174,179)(175,178);
s3 := Sym(180)!( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5, 10)( 16, 21)( 17, 22)
( 18, 23)( 19, 24)( 20, 25)( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)
( 46, 51)( 47, 52)( 48, 53)( 49, 54)( 50, 55)( 61, 66)( 62, 67)( 63, 68)
( 64, 69)( 65, 70)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)( 91, 96)
( 92, 97)( 93, 98)( 94, 99)( 95,100)(106,111)(107,112)(108,113)(109,114)
(110,115)(121,126)(122,127)(123,128)(124,129)(125,130)(136,141)(137,142)
(138,143)(139,144)(140,145)(151,156)(152,157)(153,158)(154,159)(155,160)
(166,171)(167,172)(168,173)(169,174)(170,175);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
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,
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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