Polytope of Type {4,30,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,30,6}*1440a
if this polytope has a name.
Group : SmallGroup(1440,5358)
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}*720a
   3-fold quotients : {4,10,6}*480
   5-fold quotients : {4,6,6}*288c
   6-fold quotients : {2,10,6}*240
   9-fold quotients : {4,10,2}*160
   10-fold quotients : {4,6,3}*144, {2,6,6}*144b
   15-fold quotients : {4,2,6}*96
   18-fold quotients : {2,10,2}*80
   20-fold quotients : {2,6,3}*72
   30-fold quotients : {4,2,3}*48, {2,2,6}*48
   36-fold quotients : {2,5,2}*40
   45-fold quotients : {4,2,2}*32
   60-fold quotients : {2,2,3}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

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

References : None.
to this polytope