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