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Polytope of Type {20,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,6,6}*1440a
Also Known As : {{20,6|2},{6,6|2}}. if this polytope has another name.
Group : SmallGroup(1440,5284)
Rank : 4
Schlafli Type : {20,6,6}
Number of vertices, edges, etc : 20, 60, 18, 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 : {10,6,6}*720a
   3-fold quotients : {20,2,6}*480, {20,6,2}*480a
   5-fold quotients : {4,6,6}*288a
   6-fold quotients : {20,2,3}*240, {10,2,6}*240, {10,6,2}*240
   9-fold quotients : {20,2,2}*160
   10-fold quotients : {2,6,6}*144a
   12-fold quotients : {5,2,6}*120, {10,2,3}*120
   15-fold quotients : {4,2,6}*96, {4,6,2}*96a
   18-fold quotients : {10,2,2}*80
   24-fold quotients : {5,2,3}*60
   30-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
   36-fold quotients : {5,2,2}*40
   45-fold quotients : {4,2,2}*32
   60-fold quotients : {2,2,3}*24, {2,3,2}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)( 78, 79)
( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 91,136)( 92,140)( 93,139)( 94,138)
( 95,137)( 96,141)( 97,145)( 98,144)( 99,143)(100,142)(101,146)(102,150)
(103,149)(104,148)(105,147)(106,151)(107,155)(108,154)(109,153)(110,152)
(111,156)(112,160)(113,159)(114,158)(115,157)(116,161)(117,165)(118,164)
(119,163)(120,162)(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);;
s1 := (  1, 92)(  2, 91)(  3, 95)(  4, 94)(  5, 93)(  6,102)(  7,101)(  8,105)
(  9,104)( 10,103)( 11, 97)( 12, 96)( 13,100)( 14, 99)( 15, 98)( 16,107)
( 17,106)( 18,110)( 19,109)( 20,108)( 21,117)( 22,116)( 23,120)( 24,119)
( 25,118)( 26,112)( 27,111)( 28,115)( 29,114)( 30,113)( 31,122)( 32,121)
( 33,125)( 34,124)( 35,123)( 36,132)( 37,131)( 38,135)( 39,134)( 40,133)
( 41,127)( 42,126)( 43,130)( 44,129)( 45,128)( 46,137)( 47,136)( 48,140)
( 49,139)( 50,138)( 51,147)( 52,146)( 53,150)( 54,149)( 55,148)( 56,142)
( 57,141)( 58,145)( 59,144)( 60,143)( 61,152)( 62,151)( 63,155)( 64,154)
( 65,153)( 66,162)( 67,161)( 68,165)( 69,164)( 70,163)( 71,157)( 72,156)
( 73,160)( 74,159)( 75,158)( 76,167)( 77,166)( 78,170)( 79,169)( 80,168)
( 81,177)( 82,176)( 83,180)( 84,179)( 85,178)( 86,172)( 87,171)( 88,175)
( 89,174)( 90,173);;
s2 := (  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);;
s3 := (  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)( 91,106)( 92,107)
( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)
(101,116)(102,117)(103,118)(104,119)(105,120)(136,151)(137,152)(138,153)
(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)
(147,162)(148,163)(149,164)(150,165);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(180)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)
( 58, 59)( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)
( 78, 79)( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 91,136)( 92,140)( 93,139)
( 94,138)( 95,137)( 96,141)( 97,145)( 98,144)( 99,143)(100,142)(101,146)
(102,150)(103,149)(104,148)(105,147)(106,151)(107,155)(108,154)(109,153)
(110,152)(111,156)(112,160)(113,159)(114,158)(115,157)(116,161)(117,165)
(118,164)(119,163)(120,162)(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);
s1 := Sym(180)!(  1, 92)(  2, 91)(  3, 95)(  4, 94)(  5, 93)(  6,102)(  7,101)
(  8,105)(  9,104)( 10,103)( 11, 97)( 12, 96)( 13,100)( 14, 99)( 15, 98)
( 16,107)( 17,106)( 18,110)( 19,109)( 20,108)( 21,117)( 22,116)( 23,120)
( 24,119)( 25,118)( 26,112)( 27,111)( 28,115)( 29,114)( 30,113)( 31,122)
( 32,121)( 33,125)( 34,124)( 35,123)( 36,132)( 37,131)( 38,135)( 39,134)
( 40,133)( 41,127)( 42,126)( 43,130)( 44,129)( 45,128)( 46,137)( 47,136)
( 48,140)( 49,139)( 50,138)( 51,147)( 52,146)( 53,150)( 54,149)( 55,148)
( 56,142)( 57,141)( 58,145)( 59,144)( 60,143)( 61,152)( 62,151)( 63,155)
( 64,154)( 65,153)( 66,162)( 67,161)( 68,165)( 69,164)( 70,163)( 71,157)
( 72,156)( 73,160)( 74,159)( 75,158)( 76,167)( 77,166)( 78,170)( 79,169)
( 80,168)( 81,177)( 82,176)( 83,180)( 84,179)( 85,178)( 86,172)( 87,171)
( 88,175)( 89,174)( 90,173);
s2 := 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);
s3 := 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)( 91,106)
( 92,107)( 93,108)( 94,109)( 95,110)( 96,111)( 97,112)( 98,113)( 99,114)
(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(136,151)(137,152)
(138,153)(139,154)(140,155)(141,156)(142,157)(143,158)(144,159)(145,160)
(146,161)(147,162)(148,163)(149,164)(150,165);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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