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Polytope of Type {6,6,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,20}*1440a
Also Known As : {{6,6|2},{6,20|2}}. if this polytope has another name.
Group : SmallGroup(1440,5284)
Rank : 4
Schlafli Type : {6,6,20}
Number of vertices, edges, etc : 6, 18, 60, 20
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 : {6,6,10}*720a
   3-fold quotients : {2,6,20}*480a, {6,2,20}*480
   5-fold quotients : {6,6,4}*288a
   6-fold quotients : {3,2,20}*240, {2,6,10}*240, {6,2,10}*240
   9-fold quotients : {2,2,20}*160
   10-fold quotients : {6,6,2}*144a
   12-fold quotients : {3,2,10}*120, {6,2,5}*120
   15-fold quotients : {2,6,4}*96a, {6,2,4}*96
   18-fold quotients : {2,2,10}*80
   24-fold quotients : {3,2,5}*60
   30-fold quotients : {3,2,4}*48, {2,6,2}*48, {6,2,2}*48
   36-fold quotients : {2,2,5}*40
   45-fold quotients : {2,2,4}*32
   60-fold quotients : {2,3,2}*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 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)
( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)
( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)( 69, 74)
( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)( 97,102)
( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)(115,120)
(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)(143,148)
(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)(171,176)
(172,177)(173,178)(174,179)(175,180);;
s1 := (  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 36)( 17, 37)( 18, 38)
( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)( 26, 41)
( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)( 49, 54)
( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)( 67, 77)
( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)
( 91, 96)( 92, 97)( 93, 98)( 94, 99)( 95,100)(106,126)(107,127)(108,128)
(109,129)(110,130)(111,121)(112,122)(113,123)(114,124)(115,125)(116,131)
(117,132)(118,133)(119,134)(120,135)(136,141)(137,142)(138,143)(139,144)
(140,145)(151,171)(152,172)(153,173)(154,174)(155,175)(156,166)(157,167)
(158,168)(159,169)(160,170)(161,176)(162,177)(163,178)(164,179)(165,180);;
s2 := (  1, 16)(  2, 20)(  3, 19)(  4, 18)(  5, 17)(  6, 21)(  7, 25)(  8, 24)
(  9, 23)( 10, 22)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)( 32, 35)
( 33, 34)( 37, 40)( 38, 39)( 42, 45)( 43, 44)( 46, 61)( 47, 65)( 48, 64)
( 49, 63)( 50, 62)( 51, 66)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 71)
( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 77, 80)( 78, 79)( 82, 85)( 83, 84)
( 87, 90)( 88, 89)( 91,151)( 92,155)( 93,154)( 94,153)( 95,152)( 96,156)
( 97,160)( 98,159)( 99,158)(100,157)(101,161)(102,165)(103,164)(104,163)
(105,162)(106,136)(107,140)(108,139)(109,138)(110,137)(111,141)(112,145)
(113,144)(114,143)(115,142)(116,146)(117,150)(118,149)(119,148)(120,147)
(121,166)(122,170)(123,169)(124,168)(125,167)(126,171)(127,175)(128,174)
(129,173)(130,172)(131,176)(132,180)(133,179)(134,178)(135,177);;
s3 := (  1, 92)(  2, 91)(  3, 95)(  4, 94)(  5, 93)(  6, 97)(  7, 96)(  8,100)
(  9, 99)( 10, 98)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)( 16,107)
( 17,106)( 18,110)( 19,109)( 20,108)( 21,112)( 22,111)( 23,115)( 24,114)
( 25,113)( 26,117)( 27,116)( 28,120)( 29,119)( 30,118)( 31,122)( 32,121)
( 33,125)( 34,124)( 35,123)( 36,127)( 37,126)( 38,130)( 39,129)( 40,128)
( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,137)( 47,136)( 48,140)
( 49,139)( 50,138)( 51,142)( 52,141)( 53,145)( 54,144)( 55,143)( 56,147)
( 57,146)( 58,150)( 59,149)( 60,148)( 61,152)( 62,151)( 63,155)( 64,154)
( 65,153)( 66,157)( 67,156)( 68,160)( 69,159)( 70,158)( 71,162)( 72,161)
( 73,165)( 74,164)( 75,163)( 76,167)( 77,166)( 78,170)( 79,169)( 80,168)
( 81,172)( 82,171)( 83,175)( 84,174)( 85,173)( 86,177)( 87,176)( 88,180)
( 89,179)( 90,178);;
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*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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)!(  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)
( 23, 28)( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)
( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)
( 69, 74)( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)
( 97,102)( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)
(115,120)(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)
(143,148)(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)
(171,176)(172,177)(173,178)(174,179)(175,180);
s1 := Sym(180)!(  1,  6)(  2,  7)(  3,  8)(  4,  9)(  5, 10)( 16, 36)( 17, 37)
( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)
( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 51)( 47, 52)( 48, 53)
( 49, 54)( 50, 55)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 85)( 66, 76)
( 67, 77)( 68, 78)( 69, 79)( 70, 80)( 71, 86)( 72, 87)( 73, 88)( 74, 89)
( 75, 90)( 91, 96)( 92, 97)( 93, 98)( 94, 99)( 95,100)(106,126)(107,127)
(108,128)(109,129)(110,130)(111,121)(112,122)(113,123)(114,124)(115,125)
(116,131)(117,132)(118,133)(119,134)(120,135)(136,141)(137,142)(138,143)
(139,144)(140,145)(151,171)(152,172)(153,173)(154,174)(155,175)(156,166)
(157,167)(158,168)(159,169)(160,170)(161,176)(162,177)(163,178)(164,179)
(165,180);
s2 := Sym(180)!(  1, 16)(  2, 20)(  3, 19)(  4, 18)(  5, 17)(  6, 21)(  7, 25)
(  8, 24)(  9, 23)( 10, 22)( 11, 26)( 12, 30)( 13, 29)( 14, 28)( 15, 27)
( 32, 35)( 33, 34)( 37, 40)( 38, 39)( 42, 45)( 43, 44)( 46, 61)( 47, 65)
( 48, 64)( 49, 63)( 50, 62)( 51, 66)( 52, 70)( 53, 69)( 54, 68)( 55, 67)
( 56, 71)( 57, 75)( 58, 74)( 59, 73)( 60, 72)( 77, 80)( 78, 79)( 82, 85)
( 83, 84)( 87, 90)( 88, 89)( 91,151)( 92,155)( 93,154)( 94,153)( 95,152)
( 96,156)( 97,160)( 98,159)( 99,158)(100,157)(101,161)(102,165)(103,164)
(104,163)(105,162)(106,136)(107,140)(108,139)(109,138)(110,137)(111,141)
(112,145)(113,144)(114,143)(115,142)(116,146)(117,150)(118,149)(119,148)
(120,147)(121,166)(122,170)(123,169)(124,168)(125,167)(126,171)(127,175)
(128,174)(129,173)(130,172)(131,176)(132,180)(133,179)(134,178)(135,177);
s3 := Sym(180)!(  1, 92)(  2, 91)(  3, 95)(  4, 94)(  5, 93)(  6, 97)(  7, 96)
(  8,100)(  9, 99)( 10, 98)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)
( 16,107)( 17,106)( 18,110)( 19,109)( 20,108)( 21,112)( 22,111)( 23,115)
( 24,114)( 25,113)( 26,117)( 27,116)( 28,120)( 29,119)( 30,118)( 31,122)
( 32,121)( 33,125)( 34,124)( 35,123)( 36,127)( 37,126)( 38,130)( 39,129)
( 40,128)( 41,132)( 42,131)( 43,135)( 44,134)( 45,133)( 46,137)( 47,136)
( 48,140)( 49,139)( 50,138)( 51,142)( 52,141)( 53,145)( 54,144)( 55,143)
( 56,147)( 57,146)( 58,150)( 59,149)( 60,148)( 61,152)( 62,151)( 63,155)
( 64,154)( 65,153)( 66,157)( 67,156)( 68,160)( 69,159)( 70,158)( 71,162)
( 72,161)( 73,165)( 74,164)( 75,163)( 76,167)( 77,166)( 78,170)( 79,169)
( 80,168)( 81,172)( 82,171)( 83,175)( 84,174)( 85,173)( 86,177)( 87,176)
( 88,180)( 89,179)( 90,178);
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*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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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