Polytope of Type {4,20,6}

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

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