Polytope of Type {6,12,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,10}*1440a
Also Known As : {{6,12|2},{12,10|2}}. if this polytope has another 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}*720a
3-fold quotients : {2,12,10}*480, {6,4,10}*480
5-fold quotients : {6,12,2}*288a
6-fold quotients : {2,6,10}*240, {6,2,10}*240
9-fold quotients : {2,4,10}*160
10-fold quotients : {6,6,2}*144a
12-fold quotients : {3,2,10}*120, {6,2,5}*120
15-fold quotients : {2,12,2}*96, {6,4,2}*96a
18-fold quotients : {2,2,10}*80
24-fold quotients : {3,2,5}*60
30-fold quotients : {2,6,2}*48, {6,2,2}*48
36-fold quotients : {2,2,5}*40
45-fold quotients : {2,4,2}*32
60-fold quotients : {2,3,2}*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 := ( 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)( 6,111)( 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);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, 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)( 6,111)( 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);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
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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