Polytope of Type {6,20,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,20,4}*1920a
if this polytope has a name.
Group : SmallGroup(1920,151308)
Rank : 4
Schlafli Type : {6,20,4}
Number of vertices, edges, etc : 6, 120, 80, 8
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 : {6,20,4}*960
   3-fold quotients : {2,20,4}*640
   4-fold quotients : {6,20,2}*480a, {6,10,4}*480
   5-fold quotients : {6,4,4}*384a
   6-fold quotients : {2,20,4}*320
   8-fold quotients : {6,10,2}*240
   10-fold quotients : {6,4,4}*192
   12-fold quotients : {2,20,2}*160, {2,10,4}*160
   15-fold quotients : {2,4,4}*128
   20-fold quotients : {6,2,4}*96, {6,4,2}*96a
   24-fold quotients : {2,10,2}*80
   30-fold quotients : {2,4,4}*64
   40-fold quotients : {3,2,4}*48, {6,2,2}*48
   48-fold quotients : {2,5,2}*40
   60-fold quotients : {2,2,4}*32, {2,4,2}*32
   80-fold quotients : {3,2,2}*24
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope