Polytope of Type {10,12,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,12,6}*1440c
if this polytope has a name.
Group : SmallGroup(1440,5358)
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}*720b
3-fold quotients : {10,4,6}*480
4-fold quotients : {10,6,3}*360
5-fold quotients : {2,12,6}*288c
6-fold quotients : {10,2,6}*240
9-fold quotients : {10,4,2}*160
10-fold quotients : {2,6,6}*144b
12-fold quotients : {5,2,6}*120, {10,2,3}*120
15-fold quotients : {2,4,6}*96a
18-fold quotients : {10,2,2}*80
20-fold quotients : {2,6,3}*72
24-fold quotients : {5,2,3}*60
30-fold quotients : {2,2,6}*48
36-fold quotients : {5,2,2}*40
45-fold quotients : {2,4,2}*32
60-fold quotients : {2,2,3}*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, 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)( 91,137)( 92,136)( 93,140)( 94,139)( 95,138)( 96,147)
( 97,146)( 98,150)( 99,149)(100,148)(101,142)(102,141)(103,145)(104,144)
(105,143)(106,152)(107,151)(108,155)(109,154)(110,153)(111,162)(112,161)
(113,165)(114,164)(115,163)(116,157)(117,156)(118,160)(119,159)(120,158)
(121,167)(122,166)(123,170)(124,169)(125,168)(126,177)(127,176)(128,180)
(129,179)(130,178)(131,172)(132,171)(133,175)(134,174)(135,173);;
s2 := ( 1, 96)( 2, 97)( 3, 98)( 4, 99)( 5,100)( 6, 91)( 7, 92)( 8, 93)
( 9, 94)( 10, 95)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,126)
( 17,127)( 18,128)( 19,129)( 20,130)( 21,121)( 22,122)( 23,123)( 24,124)
( 25,125)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)
( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46,141)( 47,142)( 48,143)
( 49,144)( 50,145)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)( 64,174)
( 65,175)( 66,166)( 67,167)( 68,168)( 69,169)( 70,170)( 71,176)( 72,177)
( 73,178)( 74,179)( 75,180)( 76,156)( 77,157)( 78,158)( 79,159)( 80,160)
( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,161)( 87,162)( 88,163)
( 89,164)( 90,165);;
s3 := ( 1, 16)( 2, 17)( 3, 18)( 4, 19)( 5, 20)( 6, 26)( 7, 27)( 8, 28)
( 9, 29)( 10, 30)( 11, 21)( 12, 22)( 13, 23)( 14, 24)( 15, 25)( 36, 41)
( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 46, 61)( 47, 62)( 48, 63)( 49, 64)
( 50, 65)( 51, 71)( 52, 72)( 53, 73)( 54, 74)( 55, 75)( 56, 66)( 57, 67)
( 58, 68)( 59, 69)( 60, 70)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)
( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)( 96,116)( 97,117)( 98,118)
( 99,119)(100,120)(101,111)(102,112)(103,113)(104,114)(105,115)(126,131)
(127,132)(128,133)(129,134)(130,135)(136,151)(137,152)(138,153)(139,154)
(140,155)(141,161)(142,162)(143,163)(144,164)(145,165)(146,156)(147,157)
(148,158)(149,159)(150,160)(171,176)(172,177)(173,178)(174,179)(175,180);;
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,
s3*s1*s2*s1*s2*s3*s1*s2*s1*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 ];;
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, 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)( 91,137)( 92,136)( 93,140)( 94,139)( 95,138)
( 96,147)( 97,146)( 98,150)( 99,149)(100,148)(101,142)(102,141)(103,145)
(104,144)(105,143)(106,152)(107,151)(108,155)(109,154)(110,153)(111,162)
(112,161)(113,165)(114,164)(115,163)(116,157)(117,156)(118,160)(119,159)
(120,158)(121,167)(122,166)(123,170)(124,169)(125,168)(126,177)(127,176)
(128,180)(129,179)(130,178)(131,172)(132,171)(133,175)(134,174)(135,173);
s2 := Sym(180)!( 1, 96)( 2, 97)( 3, 98)( 4, 99)( 5,100)( 6, 91)( 7, 92)
( 8, 93)( 9, 94)( 10, 95)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)
( 16,126)( 17,127)( 18,128)( 19,129)( 20,130)( 21,121)( 22,122)( 23,123)
( 24,124)( 25,125)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,111)
( 32,112)( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)
( 40,110)( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46,141)( 47,142)
( 48,143)( 49,144)( 50,145)( 51,136)( 52,137)( 53,138)( 54,139)( 55,140)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,171)( 62,172)( 63,173)
( 64,174)( 65,175)( 66,166)( 67,167)( 68,168)( 69,169)( 70,170)( 71,176)
( 72,177)( 73,178)( 74,179)( 75,180)( 76,156)( 77,157)( 78,158)( 79,159)
( 80,160)( 81,151)( 82,152)( 83,153)( 84,154)( 85,155)( 86,161)( 87,162)
( 88,163)( 89,164)( 90,165);
s3 := Sym(180)!( 1, 16)( 2, 17)( 3, 18)( 4, 19)( 5, 20)( 6, 26)( 7, 27)
( 8, 28)( 9, 29)( 10, 30)( 11, 21)( 12, 22)( 13, 23)( 14, 24)( 15, 25)
( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 46, 61)( 47, 62)( 48, 63)
( 49, 64)( 50, 65)( 51, 71)( 52, 72)( 53, 73)( 54, 74)( 55, 75)( 56, 66)
( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 81, 86)( 82, 87)( 83, 88)( 84, 89)
( 85, 90)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)( 96,116)( 97,117)
( 98,118)( 99,119)(100,120)(101,111)(102,112)(103,113)(104,114)(105,115)
(126,131)(127,132)(128,133)(129,134)(130,135)(136,151)(137,152)(138,153)
(139,154)(140,155)(141,161)(142,162)(143,163)(144,164)(145,165)(146,156)
(147,157)(148,158)(149,159)(150,160)(171,176)(172,177)(173,178)(174,179)
(175,180);
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, s3*s1*s2*s1*s2*s3*s1*s2*s1*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 >;
References : None.
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