Polytope of Type {12,6,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,10}*1440a
Also Known As : {{12,6|2},{6,10|2}}. if this polytope has another name.
Group : SmallGroup(1440,5282)
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}*720a
3-fold quotients : {12,2,10}*480, {4,6,10}*480a
5-fold quotients : {12,6,2}*288a
6-fold quotients : {12,2,5}*240, {2,6,10}*240, {6,2,10}*240
9-fold quotients : {4,2,10}*160
10-fold quotients : {6,6,2}*144a
12-fold quotients : {3,2,10}*120, {6,2,5}*120
15-fold quotients : {12,2,2}*96, {4,6,2}*96a
18-fold quotients : {4,2,5}*80, {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 : {4,2,2}*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 := ( 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)( 2,107)( 3,108)( 4,109)( 5,110)( 6,116)( 7,117)( 8,118)
( 9,119)( 10,120)( 11,111)( 12,112)( 13,113)( 14,114)( 15,115)( 16, 91)
( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21,101)( 22,102)( 23,103)( 24,104)
( 25,105)( 26, 96)( 27, 97)( 28, 98)( 29, 99)( 30,100)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,131)( 37,132)( 38,133)( 39,134)( 40,135)
( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,151)( 47,152)( 48,153)
( 49,154)( 50,155)( 51,161)( 52,162)( 53,163)( 54,164)( 55,165)( 56,156)
( 57,157)( 58,158)( 59,159)( 60,160)( 61,136)( 62,137)( 63,138)( 64,139)
( 65,140)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,141)( 72,142)
( 73,143)( 74,144)( 75,145)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,176)( 82,177)( 83,178)( 84,179)( 85,180)( 86,171)( 87,172)( 88,173)
( 89,174)( 90,175);;
s2 := ( 1, 6)( 2, 10)( 3, 9)( 4, 8)( 5, 7)( 12, 15)( 13, 14)( 16, 21)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)( 48, 54)
( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)( 64, 68)
( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)
( 87, 90)( 88, 89)( 91, 96)( 92,100)( 93, 99)( 94, 98)( 95, 97)(102,105)
(103,104)(106,111)(107,115)(108,114)(109,113)(110,112)(117,120)(118,119)
(121,126)(122,130)(123,129)(124,128)(125,127)(132,135)(133,134)(136,141)
(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,156)(152,160)
(153,159)(154,158)(155,157)(162,165)(163,164)(166,171)(167,175)(168,174)
(169,173)(170,172)(177,180)(178,179);;
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, 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,
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)( 2,107)( 3,108)( 4,109)( 5,110)( 6,116)( 7,117)
( 8,118)( 9,119)( 10,120)( 11,111)( 12,112)( 13,113)( 14,114)( 15,115)
( 16, 91)( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21,101)( 22,102)( 23,103)
( 24,104)( 25,105)( 26, 96)( 27, 97)( 28, 98)( 29, 99)( 30,100)( 31,121)
( 32,122)( 33,123)( 34,124)( 35,125)( 36,131)( 37,132)( 38,133)( 39,134)
( 40,135)( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,151)( 47,152)
( 48,153)( 49,154)( 50,155)( 51,161)( 52,162)( 53,163)( 54,164)( 55,165)
( 56,156)( 57,157)( 58,158)( 59,159)( 60,160)( 61,136)( 62,137)( 63,138)
( 64,139)( 65,140)( 66,146)( 67,147)( 68,148)( 69,149)( 70,150)( 71,141)
( 72,142)( 73,143)( 74,144)( 75,145)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,176)( 82,177)( 83,178)( 84,179)( 85,180)( 86,171)( 87,172)
( 88,173)( 89,174)( 90,175);
s2 := Sym(180)!( 1, 6)( 2, 10)( 3, 9)( 4, 8)( 5, 7)( 12, 15)( 13, 14)
( 16, 21)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)
( 48, 54)( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)
( 64, 68)( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)
( 80, 82)( 87, 90)( 88, 89)( 91, 96)( 92,100)( 93, 99)( 94, 98)( 95, 97)
(102,105)(103,104)(106,111)(107,115)(108,114)(109,113)(110,112)(117,120)
(118,119)(121,126)(122,130)(123,129)(124,128)(125,127)(132,135)(133,134)
(136,141)(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,156)
(152,160)(153,159)(154,158)(155,157)(162,165)(163,164)(166,171)(167,175)
(168,174)(169,173)(170,172)(177,180)(178,179);
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,
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,
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.
to this polytope