Polytope of Type {4,30,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,30,6}*1440c
if this polytope has a 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}*720c
   3-fold quotients : {4,30,2}*480a
   4-fold quotients : {2,15,6}*360
   5-fold quotients : {4,6,6}*288b
   6-fold quotients : {2,30,2}*240
   9-fold quotients : {4,10,2}*160
   10-fold quotients : {2,6,6}*144c
   12-fold quotients : {2,15,2}*120
   15-fold quotients : {4,6,2}*96a
   18-fold quotients : {2,10,2}*80
   20-fold quotients : {2,3,6}*72
   30-fold quotients : {2,6,2}*48
   36-fold quotients : {2,5,2}*40
   45-fold quotients : {4,2,2}*32
   60-fold quotients : {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)( 16,121)
( 17,125)( 18,124)( 19,123)( 20,122)( 21,131)( 22,135)( 23,134)( 24,133)
( 25,132)( 26,126)( 27,130)( 28,129)( 29,128)( 30,127)( 31,106)( 32,110)
( 33,109)( 34,108)( 35,107)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)
( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 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)( 61,166)( 62,170)( 63,169)( 64,168)
( 65,167)( 66,176)( 67,180)( 68,179)( 69,178)( 70,177)( 71,171)( 72,175)
( 73,174)( 74,173)( 75,172)( 76,151)( 77,155)( 78,154)( 79,153)( 80,152)
( 81,161)( 82,165)( 83,164)( 84,163)( 85,162)( 86,156)( 87,160)( 88,159)
( 89,158)( 90,157);;
s2 := (  1, 22)(  2, 21)(  3, 25)(  4, 24)(  5, 23)(  6, 17)(  7, 16)(  8, 20)
(  9, 19)( 10, 18)( 11, 27)( 12, 26)( 13, 30)( 14, 29)( 15, 28)( 31, 37)
( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 67)( 47, 66)
( 48, 70)( 49, 69)( 50, 68)( 51, 62)( 52, 61)( 53, 65)( 54, 64)( 55, 63)
( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)( 76, 82)( 77, 81)( 78, 85)
( 79, 84)( 80, 83)( 86, 87)( 88, 90)( 91,112)( 92,111)( 93,115)( 94,114)
( 95,113)( 96,107)( 97,106)( 98,110)( 99,109)(100,108)(101,117)(102,116)
(103,120)(104,119)(105,118)(121,127)(122,126)(123,130)(124,129)(125,128)
(131,132)(133,135)(136,157)(137,156)(138,160)(139,159)(140,158)(141,152)
(142,151)(143,155)(144,154)(145,153)(146,162)(147,161)(148,165)(149,164)
(150,163)(166,172)(167,171)(168,175)(169,174)(170,173)(176,177)(178,180);;
s3 := ( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)
( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 61, 76)
( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)
( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)(107,122)
(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)
(116,131)(117,132)(118,133)(119,134)(120,135)(151,166)(152,167)(153,168)
(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)
(162,177)(163,178)(164,179)(165,180);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*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)
( 16,121)( 17,125)( 18,124)( 19,123)( 20,122)( 21,131)( 22,135)( 23,134)
( 24,133)( 25,132)( 26,126)( 27,130)( 28,129)( 29,128)( 30,127)( 31,106)
( 32,110)( 33,109)( 34,108)( 35,107)( 36,116)( 37,120)( 38,119)( 39,118)
( 40,117)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 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)( 61,166)( 62,170)( 63,169)
( 64,168)( 65,167)( 66,176)( 67,180)( 68,179)( 69,178)( 70,177)( 71,171)
( 72,175)( 73,174)( 74,173)( 75,172)( 76,151)( 77,155)( 78,154)( 79,153)
( 80,152)( 81,161)( 82,165)( 83,164)( 84,163)( 85,162)( 86,156)( 87,160)
( 88,159)( 89,158)( 90,157);
s2 := Sym(180)!(  1, 22)(  2, 21)(  3, 25)(  4, 24)(  5, 23)(  6, 17)(  7, 16)
(  8, 20)(  9, 19)( 10, 18)( 11, 27)( 12, 26)( 13, 30)( 14, 29)( 15, 28)
( 31, 37)( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 67)
( 47, 66)( 48, 70)( 49, 69)( 50, 68)( 51, 62)( 52, 61)( 53, 65)( 54, 64)
( 55, 63)( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)( 76, 82)( 77, 81)
( 78, 85)( 79, 84)( 80, 83)( 86, 87)( 88, 90)( 91,112)( 92,111)( 93,115)
( 94,114)( 95,113)( 96,107)( 97,106)( 98,110)( 99,109)(100,108)(101,117)
(102,116)(103,120)(104,119)(105,118)(121,127)(122,126)(123,130)(124,129)
(125,128)(131,132)(133,135)(136,157)(137,156)(138,160)(139,159)(140,158)
(141,152)(142,151)(143,155)(144,154)(145,153)(146,162)(147,161)(148,165)
(149,164)(150,163)(166,172)(167,171)(168,175)(169,174)(170,173)(176,177)
(178,180);
s3 := Sym(180)!( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)
( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)
( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)
( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)
(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)
(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)(151,166)(152,167)
(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)
(161,176)(162,177)(163,178)(164,179)(165,180);
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2 >;

References : None.
to this polytope