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# Polytope of Type {4,30,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,30,6}*1440b
Also Known As : {{4,30|2},{30,6|2}}. if this polytope has another name.
Group : SmallGroup(1440,5685)
Rank : 4
Schlafli Type : {4,30,6}
Number of vertices, edges, etc : 4, 60, 90, 6
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 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 : {2,30,6}*720b
3-fold quotients : {4,10,6}*480, {4,30,2}*480a
5-fold quotients : {4,6,6}*288a
6-fold quotients : {2,10,6}*240, {2,30,2}*240
9-fold quotients : {4,10,2}*160
10-fold quotients : {2,6,6}*144a
12-fold quotients : {2,15,2}*120
15-fold quotients : {4,2,6}*96, {4,6,2}*96a
18-fold quotients : {2,10,2}*80
30-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
36-fold quotients : {2,5,2}*40
45-fold quotients : {4,2,2}*32
60-fold quotients : {2,2,3}*24, {2,3,2}*24
90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)
( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,151)
(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)
(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)
(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)
(131,176)(132,177)(133,178)(134,179)(135,180);;
s1 := (  1, 91)(  2, 95)(  3, 94)(  4, 93)(  5, 92)(  6,101)(  7,105)(  8,104)
(  9,103)( 10,102)( 11, 96)( 12,100)( 13, 99)( 14, 98)( 15, 97)( 16,106)
( 17,110)( 18,109)( 19,108)( 20,107)( 21,116)( 22,120)( 23,119)( 24,118)
( 25,117)( 26,111)( 27,115)( 28,114)( 29,113)( 30,112)( 31,121)( 32,125)
( 33,124)( 34,123)( 35,122)( 36,131)( 37,135)( 38,134)( 39,133)( 40,132)
( 41,126)( 42,130)( 43,129)( 44,128)( 45,127)( 46,136)( 47,140)( 48,139)
( 49,138)( 50,137)( 51,146)( 52,150)( 53,149)( 54,148)( 55,147)( 56,141)
( 57,145)( 58,144)( 59,143)( 60,142)( 61,151)( 62,155)( 63,154)( 64,153)
( 65,152)( 66,161)( 67,165)( 68,164)( 69,163)( 70,162)( 71,156)( 72,160)
( 73,159)( 74,158)( 75,157)( 76,166)( 77,170)( 78,169)( 79,168)( 80,167)
( 81,176)( 82,180)( 83,179)( 84,178)( 85,177)( 86,171)( 87,175)( 88,174)
( 89,173)( 90,172);;
s2 := (  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 37)
( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)( 24, 34)
( 25, 33)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 52)( 47, 51)
( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 82)( 62, 81)( 63, 85)
( 64, 84)( 65, 83)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)( 71, 87)
( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 97)( 92, 96)( 93,100)( 94, 99)
( 95, 98)(101,102)(103,105)(106,127)(107,126)(108,130)(109,129)(110,128)
(111,122)(112,121)(113,125)(114,124)(115,123)(116,132)(117,131)(118,135)
(119,134)(120,133)(136,142)(137,141)(138,145)(139,144)(140,143)(146,147)
(148,150)(151,172)(152,171)(153,175)(154,174)(155,173)(156,167)(157,166)
(158,170)(159,169)(160,168)(161,177)(162,176)(163,180)(164,179)(165,178);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
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*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)!( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)
( 98,143)( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)
(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)
(122,167)(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)
(130,175)(131,176)(132,177)(133,178)(134,179)(135,180);
s1 := Sym(180)!(  1, 91)(  2, 95)(  3, 94)(  4, 93)(  5, 92)(  6,101)(  7,105)
(  8,104)(  9,103)( 10,102)( 11, 96)( 12,100)( 13, 99)( 14, 98)( 15, 97)
( 16,106)( 17,110)( 18,109)( 19,108)( 20,107)( 21,116)( 22,120)( 23,119)
( 24,118)( 25,117)( 26,111)( 27,115)( 28,114)( 29,113)( 30,112)( 31,121)
( 32,125)( 33,124)( 34,123)( 35,122)( 36,131)( 37,135)( 38,134)( 39,133)
( 40,132)( 41,126)( 42,130)( 43,129)( 44,128)( 45,127)( 46,136)( 47,140)
( 48,139)( 49,138)( 50,137)( 51,146)( 52,150)( 53,149)( 54,148)( 55,147)
( 56,141)( 57,145)( 58,144)( 59,143)( 60,142)( 61,151)( 62,155)( 63,154)
( 64,153)( 65,152)( 66,161)( 67,165)( 68,164)( 69,163)( 70,162)( 71,156)
( 72,160)( 73,159)( 74,158)( 75,157)( 76,166)( 77,170)( 78,169)( 79,168)
( 80,167)( 81,176)( 82,180)( 83,179)( 84,178)( 85,177)( 86,171)( 87,175)
( 88,174)( 89,173)( 90,172);
s2 := Sym(180)!(  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)
( 16, 37)( 17, 36)( 18, 40)( 19, 39)( 20, 38)( 21, 32)( 22, 31)( 23, 35)
( 24, 34)( 25, 33)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 52)
( 47, 51)( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 82)( 62, 81)
( 63, 85)( 64, 84)( 65, 83)( 66, 77)( 67, 76)( 68, 80)( 69, 79)( 70, 78)
( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 97)( 92, 96)( 93,100)
( 94, 99)( 95, 98)(101,102)(103,105)(106,127)(107,126)(108,130)(109,129)
(110,128)(111,122)(112,121)(113,125)(114,124)(115,123)(116,132)(117,131)
(118,135)(119,134)(120,133)(136,142)(137,141)(138,145)(139,144)(140,143)
(146,147)(148,150)(151,172)(152,171)(153,175)(154,174)(155,173)(156,167)
(157,166)(158,170)(159,169)(160,168)(161,177)(162,176)(163,180)(164,179)
(165,178);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, 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*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