Polytope of Type {3,6,4,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4,10}*1440
if this polytope has a name.
Group : SmallGroup(1440,5358)
Rank : 5
Schlafli Type : {3,6,4,10}
Number of vertices, edges, etc : 3, 9, 12, 20, 10
Order of s0s1s2s3s4 : 60
Order of s0s1s2s3s4s3s2s1 : 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 : {3,6,2,10}*720
3-fold quotients : {3,2,4,10}*480
4-fold quotients : {3,6,2,5}*360
5-fold quotients : {3,6,4,2}*288
6-fold quotients : {3,2,2,10}*240
10-fold quotients : {3,6,2,2}*144
12-fold quotients : {3,2,2,5}*120
15-fold quotients : {3,2,4,2}*96
30-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 6, 11)( 7, 12)( 8, 13)( 9, 14)( 10, 15)( 16, 31)( 17, 32)( 18, 33)
( 19, 34)( 20, 35)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)( 26, 36)
( 27, 37)( 28, 38)( 29, 39)( 30, 40)( 51, 56)( 52, 57)( 53, 58)( 54, 59)
( 55, 60)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 86)( 67, 87)
( 68, 88)( 69, 89)( 70, 90)( 71, 81)( 72, 82)( 73, 83)( 74, 84)( 75, 85)
( 96,101)( 97,102)( 98,103)( 99,104)(100,105)(106,121)(107,122)(108,123)
(109,124)(110,125)(111,131)(112,132)(113,133)(114,134)(115,135)(116,126)
(117,127)(118,128)(119,129)(120,130)(141,146)(142,147)(143,148)(144,149)
(145,150)(151,166)(152,167)(153,168)(154,169)(155,170)(156,176)(157,177)
(158,178)(159,179)(160,180)(161,171)(162,172)(163,173)(164,174)(165,175);;
s1 := ( 1, 21)( 2, 22)( 3, 23)( 4, 24)( 5, 25)( 6, 16)( 7, 17)( 8, 18)
( 9, 19)( 10, 20)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 31, 36)
( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 46, 66)( 47, 67)( 48, 68)( 49, 69)
( 50, 70)( 51, 61)( 52, 62)( 53, 63)( 54, 64)( 55, 65)( 56, 71)( 57, 72)
( 58, 73)( 59, 74)( 60, 75)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)
( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)( 96,106)( 97,107)( 98,108)
( 99,109)(100,110)(101,116)(102,117)(103,118)(104,119)(105,120)(121,126)
(122,127)(123,128)(124,129)(125,130)(136,156)(137,157)(138,158)(139,159)
(140,160)(141,151)(142,152)(143,153)(144,154)(145,155)(146,161)(147,162)
(148,163)(149,164)(150,165)(166,171)(167,172)(168,173)(169,174)(170,175);;
s2 := ( 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)( 91,136)( 92,137)
( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)( 98,148)( 99,149)(100,150)
(101,141)(102,142)(103,143)(104,144)(105,145)(106,151)(107,152)(108,153)
(109,154)(110,155)(111,161)(112,162)(113,163)(114,164)(115,165)(116,156)
(117,157)(118,158)(119,159)(120,160)(121,166)(122,167)(123,168)(124,169)
(125,170)(126,176)(127,177)(128,178)(129,179)(130,180)(131,171)(132,172)
(133,173)(134,174)(135,175);;
s3 := ( 1, 91)( 2, 95)( 3, 94)( 4, 93)( 5, 92)( 6, 96)( 7,100)( 8, 99)
( 9, 98)( 10, 97)( 11,101)( 12,105)( 13,104)( 14,103)( 15,102)( 16,106)
( 17,110)( 18,109)( 19,108)( 20,107)( 21,111)( 22,115)( 23,114)( 24,113)
( 25,112)( 26,116)( 27,120)( 28,119)( 29,118)( 30,117)( 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)( 46,136)( 47,140)( 48,139)
( 49,138)( 50,137)( 51,141)( 52,145)( 53,144)( 54,143)( 55,142)( 56,146)
( 57,150)( 58,149)( 59,148)( 60,147)( 61,151)( 62,155)( 63,154)( 64,153)
( 65,152)( 66,156)( 67,160)( 68,159)( 69,158)( 70,157)( 71,161)( 72,165)
( 73,164)( 74,163)( 75,162)( 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);;
s4 := ( 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,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(180)!( 6, 11)( 7, 12)( 8, 13)( 9, 14)( 10, 15)( 16, 31)( 17, 32)
( 18, 33)( 19, 34)( 20, 35)( 21, 41)( 22, 42)( 23, 43)( 24, 44)( 25, 45)
( 26, 36)( 27, 37)( 28, 38)( 29, 39)( 30, 40)( 51, 56)( 52, 57)( 53, 58)
( 54, 59)( 55, 60)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 86)
( 67, 87)( 68, 88)( 69, 89)( 70, 90)( 71, 81)( 72, 82)( 73, 83)( 74, 84)
( 75, 85)( 96,101)( 97,102)( 98,103)( 99,104)(100,105)(106,121)(107,122)
(108,123)(109,124)(110,125)(111,131)(112,132)(113,133)(114,134)(115,135)
(116,126)(117,127)(118,128)(119,129)(120,130)(141,146)(142,147)(143,148)
(144,149)(145,150)(151,166)(152,167)(153,168)(154,169)(155,170)(156,176)
(157,177)(158,178)(159,179)(160,180)(161,171)(162,172)(163,173)(164,174)
(165,175);
s1 := Sym(180)!( 1, 21)( 2, 22)( 3, 23)( 4, 24)( 5, 25)( 6, 16)( 7, 17)
( 8, 18)( 9, 19)( 10, 20)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)
( 31, 36)( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 46, 66)( 47, 67)( 48, 68)
( 49, 69)( 50, 70)( 51, 61)( 52, 62)( 53, 63)( 54, 64)( 55, 65)( 56, 71)
( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 76, 81)( 77, 82)( 78, 83)( 79, 84)
( 80, 85)( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)( 96,106)( 97,107)
( 98,108)( 99,109)(100,110)(101,116)(102,117)(103,118)(104,119)(105,120)
(121,126)(122,127)(123,128)(124,129)(125,130)(136,156)(137,157)(138,158)
(139,159)(140,160)(141,151)(142,152)(143,153)(144,154)(145,155)(146,161)
(147,162)(148,163)(149,164)(150,165)(166,171)(167,172)(168,173)(169,174)
(170,175);
s2 := 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)( 91,136)
( 92,137)( 93,138)( 94,139)( 95,140)( 96,146)( 97,147)( 98,148)( 99,149)
(100,150)(101,141)(102,142)(103,143)(104,144)(105,145)(106,151)(107,152)
(108,153)(109,154)(110,155)(111,161)(112,162)(113,163)(114,164)(115,165)
(116,156)(117,157)(118,158)(119,159)(120,160)(121,166)(122,167)(123,168)
(124,169)(125,170)(126,176)(127,177)(128,178)(129,179)(130,180)(131,171)
(132,172)(133,173)(134,174)(135,175);
s3 := Sym(180)!( 1, 91)( 2, 95)( 3, 94)( 4, 93)( 5, 92)( 6, 96)( 7,100)
( 8, 99)( 9, 98)( 10, 97)( 11,101)( 12,105)( 13,104)( 14,103)( 15,102)
( 16,106)( 17,110)( 18,109)( 19,108)( 20,107)( 21,111)( 22,115)( 23,114)
( 24,113)( 25,112)( 26,116)( 27,120)( 28,119)( 29,118)( 30,117)( 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)( 46,136)( 47,140)
( 48,139)( 49,138)( 50,137)( 51,141)( 52,145)( 53,144)( 54,143)( 55,142)
( 56,146)( 57,150)( 58,149)( 59,148)( 60,147)( 61,151)( 62,155)( 63,154)
( 64,153)( 65,152)( 66,156)( 67,160)( 68,159)( 69,158)( 70,157)( 71,161)
( 72,165)( 73,164)( 74,163)( 75,162)( 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);
s4 := 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,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
References : None.
to this polytope