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