Polytope of Type {4,6,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,20}*1920b
if this polytope has a name.
Group : SmallGroup(1920,240151)
Rank : 4
Schlafli Type : {4,6,20}
Number of vertices, edges, etc : 4, 24, 120, 40
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 : {2,6,20}*960c
   4-fold quotients : {4,6,10}*480a, {2,6,20}*480b
   5-fold quotients : {4,6,4}*384a
   8-fold quotients : {2,6,10}*240
   10-fold quotients : {4,6,4}*192b, {2,6,4}*192
   12-fold quotients : {4,2,10}*160
   20-fold quotients : {4,6,2}*96a, {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
   24-fold quotients : {4,2,5}*80, {2,2,10}*80
   40-fold quotients : {2,3,4}*48, {2,6,2}*48
   48-fold quotients : {2,2,5}*40
   60-fold quotients : {4,2,2}*32
   80-fold quotients : {2,3,2}*24
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)
(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)
(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)
(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)
(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)
(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)
(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)
(177,237)(178,238)(179,239)(180,240);;
s1 := (  1,121)(  2,122)(  3,124)(  4,123)(  5,125)(  6,126)(  7,128)(  8,127)
(  9,129)( 10,130)( 11,132)( 12,131)( 13,133)( 14,134)( 15,136)( 16,135)
( 17,137)( 18,138)( 19,140)( 20,139)( 21,161)( 22,162)( 23,164)( 24,163)
( 25,165)( 26,166)( 27,168)( 28,167)( 29,169)( 30,170)( 31,172)( 32,171)
( 33,173)( 34,174)( 35,176)( 36,175)( 37,177)( 38,178)( 39,180)( 40,179)
( 41,141)( 42,142)( 43,144)( 44,143)( 45,145)( 46,146)( 47,148)( 48,147)
( 49,149)( 50,150)( 51,152)( 52,151)( 53,153)( 54,154)( 55,156)( 56,155)
( 57,157)( 58,158)( 59,160)( 60,159)( 61,181)( 62,182)( 63,184)( 64,183)
( 65,185)( 66,186)( 67,188)( 68,187)( 69,189)( 70,190)( 71,192)( 72,191)
( 73,193)( 74,194)( 75,196)( 76,195)( 77,197)( 78,198)( 79,200)( 80,199)
( 81,221)( 82,222)( 83,224)( 84,223)( 85,225)( 86,226)( 87,228)( 88,227)
( 89,229)( 90,230)( 91,232)( 92,231)( 93,233)( 94,234)( 95,236)( 96,235)
( 97,237)( 98,238)( 99,240)(100,239)(101,201)(102,202)(103,204)(104,203)
(105,205)(106,206)(107,208)(108,207)(109,209)(110,210)(111,212)(112,211)
(113,213)(114,214)(115,216)(116,215)(117,217)(118,218)(119,220)(120,219);;
s2 := (  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 37)(  6, 40)(  7, 39)(  8, 38)
(  9, 33)( 10, 36)( 11, 35)( 12, 34)( 13, 29)( 14, 32)( 15, 31)( 16, 30)
( 17, 25)( 18, 28)( 19, 27)( 20, 26)( 42, 44)( 45, 57)( 46, 60)( 47, 59)
( 48, 58)( 49, 53)( 50, 56)( 51, 55)( 52, 54)( 61, 81)( 62, 84)( 63, 83)
( 64, 82)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69, 93)( 70, 96)( 71, 95)
( 72, 94)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)( 79, 87)
( 80, 86)(102,104)(105,117)(106,120)(107,119)(108,118)(109,113)(110,116)
(111,115)(112,114)(121,141)(122,144)(123,143)(124,142)(125,157)(126,160)
(127,159)(128,158)(129,153)(130,156)(131,155)(132,154)(133,149)(134,152)
(135,151)(136,150)(137,145)(138,148)(139,147)(140,146)(162,164)(165,177)
(166,180)(167,179)(168,178)(169,173)(170,176)(171,175)(172,174)(181,201)
(182,204)(183,203)(184,202)(185,217)(186,220)(187,219)(188,218)(189,213)
(190,216)(191,215)(192,214)(193,209)(194,212)(195,211)(196,210)(197,205)
(198,208)(199,207)(200,206)(222,224)(225,237)(226,240)(227,239)(228,238)
(229,233)(230,236)(231,235)(232,234);;
s3 := (  1,  6)(  2,  5)(  3,  8)(  4,  7)(  9, 18)( 10, 17)( 11, 20)( 12, 19)
( 13, 14)( 15, 16)( 21, 26)( 22, 25)( 23, 28)( 24, 27)( 29, 38)( 30, 37)
( 31, 40)( 32, 39)( 33, 34)( 35, 36)( 41, 46)( 42, 45)( 43, 48)( 44, 47)
( 49, 58)( 50, 57)( 51, 60)( 52, 59)( 53, 54)( 55, 56)( 61, 66)( 62, 65)
( 63, 68)( 64, 67)( 69, 78)( 70, 77)( 71, 80)( 72, 79)( 73, 74)( 75, 76)
( 81, 86)( 82, 85)( 83, 88)( 84, 87)( 89, 98)( 90, 97)( 91,100)( 92, 99)
( 93, 94)( 95, 96)(101,106)(102,105)(103,108)(104,107)(109,118)(110,117)
(111,120)(112,119)(113,114)(115,116)(121,126)(122,125)(123,128)(124,127)
(129,138)(130,137)(131,140)(132,139)(133,134)(135,136)(141,146)(142,145)
(143,148)(144,147)(149,158)(150,157)(151,160)(152,159)(153,154)(155,156)
(161,166)(162,165)(163,168)(164,167)(169,178)(170,177)(171,180)(172,179)
(173,174)(175,176)(181,186)(182,185)(183,188)(184,187)(189,198)(190,197)
(191,200)(192,199)(193,194)(195,196)(201,206)(202,205)(203,208)(204,207)
(209,218)(210,217)(211,220)(212,219)(213,214)(215,216)(221,226)(222,225)
(223,228)(224,227)(229,238)(230,237)(231,240)(232,239)(233,234)(235,236);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(240)!(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)
(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)
(136,196)(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)
(144,204)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)
(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)
(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)
(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)
(176,236)(177,237)(178,238)(179,239)(180,240);
s1 := Sym(240)!(  1,121)(  2,122)(  3,124)(  4,123)(  5,125)(  6,126)(  7,128)
(  8,127)(  9,129)( 10,130)( 11,132)( 12,131)( 13,133)( 14,134)( 15,136)
( 16,135)( 17,137)( 18,138)( 19,140)( 20,139)( 21,161)( 22,162)( 23,164)
( 24,163)( 25,165)( 26,166)( 27,168)( 28,167)( 29,169)( 30,170)( 31,172)
( 32,171)( 33,173)( 34,174)( 35,176)( 36,175)( 37,177)( 38,178)( 39,180)
( 40,179)( 41,141)( 42,142)( 43,144)( 44,143)( 45,145)( 46,146)( 47,148)
( 48,147)( 49,149)( 50,150)( 51,152)( 52,151)( 53,153)( 54,154)( 55,156)
( 56,155)( 57,157)( 58,158)( 59,160)( 60,159)( 61,181)( 62,182)( 63,184)
( 64,183)( 65,185)( 66,186)( 67,188)( 68,187)( 69,189)( 70,190)( 71,192)
( 72,191)( 73,193)( 74,194)( 75,196)( 76,195)( 77,197)( 78,198)( 79,200)
( 80,199)( 81,221)( 82,222)( 83,224)( 84,223)( 85,225)( 86,226)( 87,228)
( 88,227)( 89,229)( 90,230)( 91,232)( 92,231)( 93,233)( 94,234)( 95,236)
( 96,235)( 97,237)( 98,238)( 99,240)(100,239)(101,201)(102,202)(103,204)
(104,203)(105,205)(106,206)(107,208)(108,207)(109,209)(110,210)(111,212)
(112,211)(113,213)(114,214)(115,216)(116,215)(117,217)(118,218)(119,220)
(120,219);
s2 := Sym(240)!(  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 37)(  6, 40)(  7, 39)
(  8, 38)(  9, 33)( 10, 36)( 11, 35)( 12, 34)( 13, 29)( 14, 32)( 15, 31)
( 16, 30)( 17, 25)( 18, 28)( 19, 27)( 20, 26)( 42, 44)( 45, 57)( 46, 60)
( 47, 59)( 48, 58)( 49, 53)( 50, 56)( 51, 55)( 52, 54)( 61, 81)( 62, 84)
( 63, 83)( 64, 82)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69, 93)( 70, 96)
( 71, 95)( 72, 94)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)
( 79, 87)( 80, 86)(102,104)(105,117)(106,120)(107,119)(108,118)(109,113)
(110,116)(111,115)(112,114)(121,141)(122,144)(123,143)(124,142)(125,157)
(126,160)(127,159)(128,158)(129,153)(130,156)(131,155)(132,154)(133,149)
(134,152)(135,151)(136,150)(137,145)(138,148)(139,147)(140,146)(162,164)
(165,177)(166,180)(167,179)(168,178)(169,173)(170,176)(171,175)(172,174)
(181,201)(182,204)(183,203)(184,202)(185,217)(186,220)(187,219)(188,218)
(189,213)(190,216)(191,215)(192,214)(193,209)(194,212)(195,211)(196,210)
(197,205)(198,208)(199,207)(200,206)(222,224)(225,237)(226,240)(227,239)
(228,238)(229,233)(230,236)(231,235)(232,234);
s3 := Sym(240)!(  1,  6)(  2,  5)(  3,  8)(  4,  7)(  9, 18)( 10, 17)( 11, 20)
( 12, 19)( 13, 14)( 15, 16)( 21, 26)( 22, 25)( 23, 28)( 24, 27)( 29, 38)
( 30, 37)( 31, 40)( 32, 39)( 33, 34)( 35, 36)( 41, 46)( 42, 45)( 43, 48)
( 44, 47)( 49, 58)( 50, 57)( 51, 60)( 52, 59)( 53, 54)( 55, 56)( 61, 66)
( 62, 65)( 63, 68)( 64, 67)( 69, 78)( 70, 77)( 71, 80)( 72, 79)( 73, 74)
( 75, 76)( 81, 86)( 82, 85)( 83, 88)( 84, 87)( 89, 98)( 90, 97)( 91,100)
( 92, 99)( 93, 94)( 95, 96)(101,106)(102,105)(103,108)(104,107)(109,118)
(110,117)(111,120)(112,119)(113,114)(115,116)(121,126)(122,125)(123,128)
(124,127)(129,138)(130,137)(131,140)(132,139)(133,134)(135,136)(141,146)
(142,145)(143,148)(144,147)(149,158)(150,157)(151,160)(152,159)(153,154)
(155,156)(161,166)(162,165)(163,168)(164,167)(169,178)(170,177)(171,180)
(172,179)(173,174)(175,176)(181,186)(182,185)(183,188)(184,187)(189,198)
(190,197)(191,200)(192,199)(193,194)(195,196)(201,206)(202,205)(203,208)
(204,207)(209,218)(210,217)(211,220)(212,219)(213,214)(215,216)(221,226)
(222,225)(223,228)(224,227)(229,238)(230,237)(231,240)(232,239)(233,234)
(235,236);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2 >;

References : None.
to this polytope