Polytope of Type {10,4,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,6,3}*1440
if this polytope has a name.
Group : SmallGroup(1440,5358)
Rank : 5
Schlafli Type : {10,4,6,3}
Number of vertices, edges, etc : 10, 20, 12, 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 : {10,2,6,3}*720
3-fold quotients : {10,4,2,3}*480
4-fold quotients : {5,2,6,3}*360
5-fold quotients : {2,4,6,3}*288
6-fold quotients : {10,2,2,3}*240
10-fold quotients : {2,2,6,3}*144
12-fold quotients : {5,2,2,3}*120
15-fold quotients : {2,4,2,3}*96
30-fold quotients : {2,2,2,3}*48
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, 92)( 2, 91)( 3, 95)( 4, 94)( 5, 93)( 6, 97)( 7, 96)( 8,100)
( 9, 99)( 10, 98)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)( 16,107)
( 17,106)( 18,110)( 19,109)( 20,108)( 21,112)( 22,111)( 23,115)( 24,114)
( 25,113)( 26,117)( 27,116)( 28,120)( 29,119)( 30,118)( 31,122)( 32,121)
( 33,125)( 34,124)( 35,123)( 36,127)( 37,126)( 38,130)( 39,129)( 40,128)
( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,137)( 47,136)( 48,140)
( 49,139)( 50,138)( 51,142)( 52,141)( 53,145)( 54,144)( 55,143)( 56,147)
( 57,146)( 58,150)( 59,149)( 60,148)( 61,152)( 62,151)( 63,155)( 64,154)
( 65,153)( 66,157)( 67,156)( 68,160)( 69,159)( 70,158)( 71,162)( 72,161)
( 73,165)( 74,164)( 75,163)( 76,167)( 77,166)( 78,170)( 79,169)( 80,168)
( 81,172)( 82,171)( 83,175)( 84,174)( 85,173)( 86,177)( 87,176)( 88,180)
( 89,179)( 90,178);;
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, 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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*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, 92)( 2, 91)( 3, 95)( 4, 94)( 5, 93)( 6, 97)( 7, 96)
( 8,100)( 9, 99)( 10, 98)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)
( 16,107)( 17,106)( 18,110)( 19,109)( 20,108)( 21,112)( 22,111)( 23,115)
( 24,114)( 25,113)( 26,117)( 27,116)( 28,120)( 29,119)( 30,118)( 31,122)
( 32,121)( 33,125)( 34,124)( 35,123)( 36,127)( 37,126)( 38,130)( 39,129)
( 40,128)( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,137)( 47,136)
( 48,140)( 49,139)( 50,138)( 51,142)( 52,141)( 53,145)( 54,144)( 55,143)
( 56,147)( 57,146)( 58,150)( 59,149)( 60,148)( 61,152)( 62,151)( 63,155)
( 64,154)( 65,153)( 66,157)( 67,156)( 68,160)( 69,159)( 70,158)( 71,162)
( 72,161)( 73,165)( 74,164)( 75,163)( 76,167)( 77,166)( 78,170)( 79,169)
( 80,168)( 81,172)( 82,171)( 83,175)( 84,174)( 85,173)( 86,177)( 87,176)
( 88,180)( 89,179)( 90,178);
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, 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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*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.
to this polytope