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Polytope of Type {6,4,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,30}*1440
Also Known As : {{6,4|2},{4,30|2}}. if this polytope has another name.
Group : SmallGroup(1440,5685)
Rank : 4
Schlafli Type : {6,4,30}
Number of vertices, edges, etc : 6, 12, 60, 30
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 : {6,2,30}*720
   3-fold quotients : {6,4,10}*480, {2,4,30}*480a
   4-fold quotients : {3,2,30}*360, {6,2,15}*360
   5-fold quotients : {6,4,6}*288
   6-fold quotients : {6,2,10}*240, {2,2,30}*240
   8-fold quotients : {3,2,15}*180
   9-fold quotients : {2,4,10}*160
   10-fold quotients : {6,2,6}*144
   12-fold quotients : {3,2,10}*120, {6,2,5}*120, {2,2,15}*120
   15-fold quotients : {2,4,6}*96a, {6,4,2}*96a
   18-fold quotients : {2,2,10}*80
   20-fold quotients : {3,2,6}*72, {6,2,3}*72
   24-fold quotients : {3,2,5}*60
   30-fold quotients : {2,2,6}*48, {6,2,2}*48
   36-fold quotients : {2,2,5}*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 := ( 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);;
s1 := (  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)( 91,151)( 92,152)
( 93,153)( 94,154)( 95,155)( 96,156)( 97,157)( 98,158)( 99,159)(100,160)
(101,161)(102,162)(103,163)(104,164)(105,165)(106,136)(107,137)(108,138)
(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)(116,146)
(117,147)(118,148)(119,149)(120,150)(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);;
s2 := (  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);;
s3 := (  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)( 16, 22)
( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 27)( 28, 30)( 31, 37)( 32, 36)
( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 52)( 47, 51)( 48, 55)
( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 67)( 62, 66)( 63, 70)( 64, 69)
( 65, 68)( 71, 72)( 73, 75)( 76, 82)( 77, 81)( 78, 85)( 79, 84)( 80, 83)
( 86, 87)( 88, 90)( 91, 97)( 92, 96)( 93,100)( 94, 99)( 95, 98)(101,102)
(103,105)(106,112)(107,111)(108,115)(109,114)(110,113)(116,117)(118,120)
(121,127)(122,126)(123,130)(124,129)(125,128)(131,132)(133,135)(136,142)
(137,141)(138,145)(139,144)(140,143)(146,147)(148,150)(151,157)(152,156)
(153,160)(154,159)(155,158)(161,162)(163,165)(166,172)(167,171)(168,175)
(169,174)(170,173)(176,177)(178,180);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, 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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
s1 := 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)( 91,151)
( 92,152)( 93,153)( 94,154)( 95,155)( 96,156)( 97,157)( 98,158)( 99,159)
(100,160)(101,161)(102,162)(103,163)(104,164)(105,165)(106,136)(107,137)
(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,145)
(116,146)(117,147)(118,148)(119,149)(120,150)(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);
s2 := 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);
s3 := Sym(180)!(  1,  7)(  2,  6)(  3, 10)(  4,  9)(  5,  8)( 11, 12)( 13, 15)
( 16, 22)( 17, 21)( 18, 25)( 19, 24)( 20, 23)( 26, 27)( 28, 30)( 31, 37)
( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 52)( 47, 51)
( 48, 55)( 49, 54)( 50, 53)( 56, 57)( 58, 60)( 61, 67)( 62, 66)( 63, 70)
( 64, 69)( 65, 68)( 71, 72)( 73, 75)( 76, 82)( 77, 81)( 78, 85)( 79, 84)
( 80, 83)( 86, 87)( 88, 90)( 91, 97)( 92, 96)( 93,100)( 94, 99)( 95, 98)
(101,102)(103,105)(106,112)(107,111)(108,115)(109,114)(110,113)(116,117)
(118,120)(121,127)(122,126)(123,130)(124,129)(125,128)(131,132)(133,135)
(136,142)(137,141)(138,145)(139,144)(140,143)(146,147)(148,150)(151,157)
(152,156)(153,160)(154,159)(155,158)(161,162)(163,165)(166,172)(167,171)
(168,175)(169,174)(170,173)(176,177)(178,180);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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