Polytope of Type {3,6,10,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,10,4}*1440
if this polytope has a name.
Group : SmallGroup(1440,5361)
Rank : 5
Schlafli Type : {3,6,10,4}
Number of vertices, edges, etc : 3, 9, 30, 20, 4
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 : {3,6,10,2}*720
3-fold quotients : {3,2,10,4}*480
5-fold quotients : {3,6,2,4}*288
6-fold quotients : {3,2,10,2}*240
10-fold quotients : {3,6,2,2}*144
12-fold quotients : {3,2,5,2}*120
15-fold quotients : {3,2,2,4}*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 := ( 2, 5)( 3, 4)( 6, 11)( 7, 15)( 8, 14)( 9, 13)( 10, 12)( 17, 20)
( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)( 33, 34)
( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)( 51, 56)
( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)( 67, 75)
( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)( 83, 89)
( 84, 88)( 85, 87)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)( 99,103)
(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)(115,117)
(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132)(137,140)
(138,139)(141,146)(142,150)(143,149)(144,148)(145,147)(152,155)(153,154)
(156,161)(157,165)(158,164)(159,163)(160,162)(167,170)(168,169)(171,176)
(172,180)(173,179)(174,178)(175,177);;
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)( 91,137)( 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)(106,152)(107,151)(108,155)(109,154)(110,153)
(111,157)(112,156)(113,160)(114,159)(115,158)(116,162)(117,161)(118,165)
(119,164)(120,163)(121,167)(122,166)(123,170)(124,169)(125,168)(126,172)
(127,171)(128,175)(129,174)(130,173)(131,177)(132,176)(133,180)(134,179)
(135,178);;
s4 := ( 1, 91)( 2, 92)( 3, 93)( 4, 94)( 5, 95)( 6, 96)( 7, 97)( 8, 98)
( 9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,106)
( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)( 24,114)
( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)
( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)( 56,146)
( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,154)
( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)( 72,162)
( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180);;
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*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)!( 2, 5)( 3, 4)( 6, 11)( 7, 15)( 8, 14)( 9, 13)( 10, 12)
( 17, 20)( 18, 19)( 21, 26)( 22, 30)( 23, 29)( 24, 28)( 25, 27)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 47, 50)( 48, 49)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 62, 65)( 63, 64)( 66, 71)
( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 77, 80)( 78, 79)( 81, 86)( 82, 90)
( 83, 89)( 84, 88)( 85, 87)( 92, 95)( 93, 94)( 96,101)( 97,105)( 98,104)
( 99,103)(100,102)(107,110)(108,109)(111,116)(112,120)(113,119)(114,118)
(115,117)(122,125)(123,124)(126,131)(127,135)(128,134)(129,133)(130,132)
(137,140)(138,139)(141,146)(142,150)(143,149)(144,148)(145,147)(152,155)
(153,154)(156,161)(157,165)(158,164)(159,163)(160,162)(167,170)(168,169)
(171,176)(172,180)(173,179)(174,178)(175,177);
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)( 91,137)( 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)(106,152)(107,151)(108,155)(109,154)
(110,153)(111,157)(112,156)(113,160)(114,159)(115,158)(116,162)(117,161)
(118,165)(119,164)(120,163)(121,167)(122,166)(123,170)(124,169)(125,168)
(126,172)(127,171)(128,175)(129,174)(130,173)(131,177)(132,176)(133,180)
(134,179)(135,178);
s4 := Sym(180)!( 1, 91)( 2, 92)( 3, 93)( 4, 94)( 5, 95)( 6, 96)( 7, 97)
( 8, 98)( 9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)
( 16,106)( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)
( 24,114)( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)
( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)
( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)
( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)
( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)
( 72,162)( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180);
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*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope