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Polytope of Type {4,6,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,20}*960a
Also Known As : {{4,6|2},{6,20|2}}. if this polytope has another name.
Group : SmallGroup(960,7430)
Rank : 4
Schlafli Type : {4,6,20}
Number of vertices, edges, etc : 4, 12, 60, 20
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,20,2} of size 1920
Vertex Figure Of :
   {2,4,6,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,20}*480a, {4,6,10}*480a
   3-fold quotients : {4,2,20}*320
   4-fold quotients : {2,6,10}*240
   5-fold quotients : {4,6,4}*192a
   6-fold quotients : {2,2,20}*160, {4,2,10}*160
   10-fold quotients : {2,6,4}*96a, {4,6,2}*96a
   12-fold quotients : {4,2,5}*80, {2,2,10}*80
   15-fold quotients : {4,2,4}*64
   20-fold quotients : {2,6,2}*48
   24-fold quotients : {2,2,5}*40
   30-fold quotients : {2,2,4}*32, {4,2,2}*32
   40-fold quotients : {2,3,2}*24
   60-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12,20}*1920a, {8,6,20}*1920, {4,6,40}*1920a
Permutation Representation (GAP) :
s0 := (  1,121)(  2,122)(  3,123)(  4,124)(  5,125)(  6,126)(  7,127)(  8,128)
(  9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)( 16,136)
( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)( 24,144)
( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)( 32,152)
( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)( 40,160)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,181)( 77,182)( 78,183)( 79,184)( 80,185)
( 81,186)( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)( 88,193)
( 89,194)( 90,195)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)( 96,231)
( 97,232)( 98,233)( 99,234)(100,235)(101,236)(102,237)(103,238)(104,239)
(105,240)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)
(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225);;
s1 := (  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)
(121,136)(122,137)(123,138)(124,139)(125,140)(126,146)(127,147)(128,148)
(129,149)(130,150)(131,141)(132,142)(133,143)(134,144)(135,145)(151,166)
(152,167)(153,168)(154,169)(155,170)(156,176)(157,177)(158,178)(159,179)
(160,180)(161,171)(162,172)(163,173)(164,174)(165,175)(181,196)(182,197)
(183,198)(184,199)(185,200)(186,206)(187,207)(188,208)(189,209)(190,210)
(191,201)(192,202)(193,203)(194,204)(195,205)(211,226)(212,227)(213,228)
(214,229)(215,230)(216,236)(217,237)(218,238)(219,239)(220,240)(221,231)
(222,232)(223,233)(224,234)(225,235);;
s2 := (  1, 66)(  2, 70)(  3, 69)(  4, 68)(  5, 67)(  6, 61)(  7, 65)(  8, 64)
(  9, 63)( 10, 62)( 11, 71)( 12, 75)( 13, 74)( 14, 73)( 15, 72)( 16, 81)
( 17, 85)( 18, 84)( 19, 83)( 20, 82)( 21, 76)( 22, 80)( 23, 79)( 24, 78)
( 25, 77)( 26, 86)( 27, 90)( 28, 89)( 29, 88)( 30, 87)( 31, 96)( 32,100)
( 33, 99)( 34, 98)( 35, 97)( 36, 91)( 37, 95)( 38, 94)( 39, 93)( 40, 92)
( 41,101)( 42,105)( 43,104)( 44,103)( 45,102)( 46,111)( 47,115)( 48,114)
( 49,113)( 50,112)( 51,106)( 52,110)( 53,109)( 54,108)( 55,107)( 56,116)
( 57,120)( 58,119)( 59,118)( 60,117)(121,201)(122,205)(123,204)(124,203)
(125,202)(126,196)(127,200)(128,199)(129,198)(130,197)(131,206)(132,210)
(133,209)(134,208)(135,207)(136,186)(137,190)(138,189)(139,188)(140,187)
(141,181)(142,185)(143,184)(144,183)(145,182)(146,191)(147,195)(148,194)
(149,193)(150,192)(151,231)(152,235)(153,234)(154,233)(155,232)(156,226)
(157,230)(158,229)(159,228)(160,227)(161,236)(162,240)(163,239)(164,238)
(165,237)(166,216)(167,220)(168,219)(169,218)(170,217)(171,211)(172,215)
(173,214)(174,213)(175,212)(176,221)(177,225)(178,224)(179,223)(180,222);;
s3 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 92)( 62, 91)( 63, 95)( 64, 94)( 65, 93)( 66, 97)( 67, 96)( 68,100)
( 69, 99)( 70, 98)( 71,102)( 72,101)( 73,105)( 74,104)( 75,103)( 76,107)
( 77,106)( 78,110)( 79,109)( 80,108)( 81,112)( 82,111)( 83,115)( 84,114)
( 85,113)( 86,117)( 87,116)( 88,120)( 89,119)( 90,118)(121,122)(123,125)
(126,127)(128,130)(131,132)(133,135)(136,137)(138,140)(141,142)(143,145)
(146,147)(148,150)(151,152)(153,155)(156,157)(158,160)(161,162)(163,165)
(166,167)(168,170)(171,172)(173,175)(176,177)(178,180)(181,212)(182,211)
(183,215)(184,214)(185,213)(186,217)(187,216)(188,220)(189,219)(190,218)
(191,222)(192,221)(193,225)(194,224)(195,223)(196,227)(197,226)(198,230)
(199,229)(200,228)(201,232)(202,231)(203,235)(204,234)(205,233)(206,237)
(207,236)(208,240)(209,239)(210,238);;
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, 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*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(240)!(  1,121)(  2,122)(  3,123)(  4,124)(  5,125)(  6,126)(  7,127)
(  8,128)(  9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)
( 16,136)( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)
( 24,144)( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)
( 32,152)( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)
( 40,160)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,181)( 77,182)( 78,183)( 79,184)
( 80,185)( 81,186)( 82,187)( 83,188)( 84,189)( 85,190)( 86,191)( 87,192)
( 88,193)( 89,194)( 90,195)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)
( 96,231)( 97,232)( 98,233)( 99,234)(100,235)(101,236)(102,237)(103,238)
(104,239)(105,240)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)
(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)
(120,225);
s1 := Sym(240)!(  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)(121,136)(122,137)(123,138)(124,139)(125,140)(126,146)(127,147)
(128,148)(129,149)(130,150)(131,141)(132,142)(133,143)(134,144)(135,145)
(151,166)(152,167)(153,168)(154,169)(155,170)(156,176)(157,177)(158,178)
(159,179)(160,180)(161,171)(162,172)(163,173)(164,174)(165,175)(181,196)
(182,197)(183,198)(184,199)(185,200)(186,206)(187,207)(188,208)(189,209)
(190,210)(191,201)(192,202)(193,203)(194,204)(195,205)(211,226)(212,227)
(213,228)(214,229)(215,230)(216,236)(217,237)(218,238)(219,239)(220,240)
(221,231)(222,232)(223,233)(224,234)(225,235);
s2 := Sym(240)!(  1, 66)(  2, 70)(  3, 69)(  4, 68)(  5, 67)(  6, 61)(  7, 65)
(  8, 64)(  9, 63)( 10, 62)( 11, 71)( 12, 75)( 13, 74)( 14, 73)( 15, 72)
( 16, 81)( 17, 85)( 18, 84)( 19, 83)( 20, 82)( 21, 76)( 22, 80)( 23, 79)
( 24, 78)( 25, 77)( 26, 86)( 27, 90)( 28, 89)( 29, 88)( 30, 87)( 31, 96)
( 32,100)( 33, 99)( 34, 98)( 35, 97)( 36, 91)( 37, 95)( 38, 94)( 39, 93)
( 40, 92)( 41,101)( 42,105)( 43,104)( 44,103)( 45,102)( 46,111)( 47,115)
( 48,114)( 49,113)( 50,112)( 51,106)( 52,110)( 53,109)( 54,108)( 55,107)
( 56,116)( 57,120)( 58,119)( 59,118)( 60,117)(121,201)(122,205)(123,204)
(124,203)(125,202)(126,196)(127,200)(128,199)(129,198)(130,197)(131,206)
(132,210)(133,209)(134,208)(135,207)(136,186)(137,190)(138,189)(139,188)
(140,187)(141,181)(142,185)(143,184)(144,183)(145,182)(146,191)(147,195)
(148,194)(149,193)(150,192)(151,231)(152,235)(153,234)(154,233)(155,232)
(156,226)(157,230)(158,229)(159,228)(160,227)(161,236)(162,240)(163,239)
(164,238)(165,237)(166,216)(167,220)(168,219)(169,218)(170,217)(171,211)
(172,215)(173,214)(174,213)(175,212)(176,221)(177,225)(178,224)(179,223)
(180,222);
s3 := Sym(240)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 92)( 62, 91)( 63, 95)( 64, 94)( 65, 93)( 66, 97)( 67, 96)
( 68,100)( 69, 99)( 70, 98)( 71,102)( 72,101)( 73,105)( 74,104)( 75,103)
( 76,107)( 77,106)( 78,110)( 79,109)( 80,108)( 81,112)( 82,111)( 83,115)
( 84,114)( 85,113)( 86,117)( 87,116)( 88,120)( 89,119)( 90,118)(121,122)
(123,125)(126,127)(128,130)(131,132)(133,135)(136,137)(138,140)(141,142)
(143,145)(146,147)(148,150)(151,152)(153,155)(156,157)(158,160)(161,162)
(163,165)(166,167)(168,170)(171,172)(173,175)(176,177)(178,180)(181,212)
(182,211)(183,215)(184,214)(185,213)(186,217)(187,216)(188,220)(189,219)
(190,218)(191,222)(192,221)(193,225)(194,224)(195,223)(196,227)(197,226)
(198,230)(199,229)(200,228)(201,232)(202,231)(203,235)(204,234)(205,233)
(206,237)(207,236)(208,240)(209,239)(210,238);
poly := sub<Sym(240)|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, 
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*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.
to this polytope