Polytope of Type {6,20,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,20,6}*1440
Also Known As : {{6,20|2},{20,6|2}}. if this polytope has another name.
Group : SmallGroup(1440,5284)
Rank : 4
Schlafli Type : {6,20,6}
Number of vertices, edges, etc : 6, 60, 60, 6
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
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 : {6,10,6}*720
3-fold quotients : {2,20,6}*480a, {6,20,2}*480a
5-fold quotients : {6,4,6}*288
6-fold quotients : {2,10,6}*240, {6,10,2}*240
9-fold quotients : {2,20,2}*160
10-fold quotients : {6,2,6}*144
15-fold quotients : {2,4,6}*96a, {6,4,2}*96a
18-fold quotients : {2,10,2}*80
20-fold quotients : {3,2,6}*72, {6,2,3}*72
30-fold quotients : {2,2,6}*48, {6,2,2}*48
36-fold quotients : {2,5,2}*40
40-fold quotients : {3,2,3}*36
45-fold quotients : {2,4,2}*32
60-fold quotients : {2,2,3}*24, {3,2,2}*24
90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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)( 96,101)( 97,102)
( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)(115,120)
(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)(143,148)
(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)(171,176)
(172,177)(173,178)(174,179)(175,180);;
s1 := ( 1, 6)( 2, 10)( 3, 9)( 4, 8)( 5, 7)( 12, 15)( 13, 14)( 16, 21)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)( 48, 54)
( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)( 64, 68)
( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)
( 87, 90)( 88, 89)( 91,141)( 92,145)( 93,144)( 94,143)( 95,142)( 96,136)
( 97,140)( 98,139)( 99,138)(100,137)(101,146)(102,150)(103,149)(104,148)
(105,147)(106,156)(107,160)(108,159)(109,158)(110,157)(111,151)(112,155)
(113,154)(114,153)(115,152)(116,161)(117,165)(118,164)(119,163)(120,162)
(121,171)(122,175)(123,174)(124,173)(125,172)(126,166)(127,170)(128,169)
(129,168)(130,167)(131,176)(132,180)(133,179)(134,178)(135,177);;
s2 := ( 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)( 16,122)
( 17,121)( 18,125)( 19,124)( 20,123)( 21,127)( 22,126)( 23,130)( 24,129)
( 25,128)( 26,132)( 27,131)( 28,135)( 29,134)( 30,133)( 31,107)( 32,106)
( 33,110)( 34,109)( 35,108)( 36,112)( 37,111)( 38,115)( 39,114)( 40,113)
( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 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)( 61,167)( 62,166)( 63,170)( 64,169)
( 65,168)( 66,172)( 67,171)( 68,175)( 69,174)( 70,173)( 71,177)( 72,176)
( 73,180)( 74,179)( 75,178)( 76,152)( 77,151)( 78,155)( 79,154)( 80,153)
( 81,157)( 82,156)( 83,160)( 84,159)( 85,158)( 86,162)( 87,161)( 88,165)
( 89,164)( 90,163);;
s3 := ( 1, 16)( 2, 17)( 3, 18)( 4, 19)( 5, 20)( 6, 21)( 7, 22)( 8, 23)
( 9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 46, 61)
( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 51, 66)( 52, 67)( 53, 68)( 54, 69)
( 55, 70)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 91,106)( 92,107)
( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)
(101,116)(102,117)(103,118)(104,119)(105,120)(136,151)(137,152)(138,153)
(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)
(147,162)(148,163)(149,164)(150,165);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!( 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)( 96,101)
( 97,102)( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)
(115,120)(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)
(143,148)(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)
(171,176)(172,177)(173,178)(174,179)(175,180);
s1 := Sym(180)!( 1, 6)( 2, 10)( 3, 9)( 4, 8)( 5, 7)( 12, 15)( 13, 14)
( 16, 21)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)
( 48, 54)( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)
( 64, 68)( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)
( 80, 82)( 87, 90)( 88, 89)( 91,141)( 92,145)( 93,144)( 94,143)( 95,142)
( 96,136)( 97,140)( 98,139)( 99,138)(100,137)(101,146)(102,150)(103,149)
(104,148)(105,147)(106,156)(107,160)(108,159)(109,158)(110,157)(111,151)
(112,155)(113,154)(114,153)(115,152)(116,161)(117,165)(118,164)(119,163)
(120,162)(121,171)(122,175)(123,174)(124,173)(125,172)(126,166)(127,170)
(128,169)(129,168)(130,167)(131,176)(132,180)(133,179)(134,178)(135,177);
s2 := 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)
( 16,122)( 17,121)( 18,125)( 19,124)( 20,123)( 21,127)( 22,126)( 23,130)
( 24,129)( 25,128)( 26,132)( 27,131)( 28,135)( 29,134)( 30,133)( 31,107)
( 32,106)( 33,110)( 34,109)( 35,108)( 36,112)( 37,111)( 38,115)( 39,114)
( 40,113)( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 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)( 61,167)( 62,166)( 63,170)
( 64,169)( 65,168)( 66,172)( 67,171)( 68,175)( 69,174)( 70,173)( 71,177)
( 72,176)( 73,180)( 74,179)( 75,178)( 76,152)( 77,151)( 78,155)( 79,154)
( 80,153)( 81,157)( 82,156)( 83,160)( 84,159)( 85,158)( 86,162)( 87,161)
( 88,165)( 89,164)( 90,163);
s3 := Sym(180)!( 1, 16)( 2, 17)( 3, 18)( 4, 19)( 5, 20)( 6, 21)( 7, 22)
( 8, 23)( 9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)
( 46, 61)( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 51, 66)( 52, 67)( 53, 68)
( 54, 69)( 55, 70)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 91,106)
( 92,107)( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)
(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(136,151)(137,152)
(138,153)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)
(146,161)(147,162)(148,163)(149,164)(150,165);
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, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
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