Polytope of Type {6,30,4}

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

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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
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, 25)(  3, 24)(  4, 23)(  5, 22)(  6, 16)(  7, 20)
(  8, 19)(  9, 18)( 10, 17)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)
( 31, 36)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 66)
( 47, 70)( 48, 69)( 49, 68)( 50, 67)( 51, 61)( 52, 65)( 53, 64)( 54, 63)
( 55, 62)( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 76, 81)( 77, 85)
( 78, 84)( 79, 83)( 80, 82)( 87, 90)( 88, 89)( 91,111)( 92,115)( 93,114)
( 94,113)( 95,112)( 96,106)( 97,110)( 98,109)( 99,108)(100,107)(101,116)
(102,120)(103,119)(104,118)(105,117)(121,126)(122,130)(123,129)(124,128)
(125,127)(132,135)(133,134)(136,156)(137,160)(138,159)(139,158)(140,157)
(141,151)(142,155)(143,154)(144,153)(145,152)(146,161)(147,165)(148,164)
(149,163)(150,162)(166,171)(167,175)(168,174)(169,173)(170,172)(177,180)
(178,179);
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)( 91,137)( 92,136)( 93,140)( 94,139)( 95,138)
( 96,147)( 97,146)( 98,150)( 99,149)(100,148)(101,142)(102,141)(103,145)
(104,144)(105,143)(106,152)(107,151)(108,155)(109,154)(110,153)(111,162)
(112,161)(113,165)(114,164)(115,163)(116,157)(117,156)(118,160)(119,159)
(120,158)(121,167)(122,166)(123,170)(124,169)(125,168)(126,177)(127,176)
(128,180)(129,179)(130,178)(131,172)(132,171)(133,175)(134,174)(135,173);
s3 := 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>;

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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 >;

References : None.
to this polytope