Polytope of Type {4,20,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20,6}*1920c
if this polytope has a name.
Group : SmallGroup(1920,240151)
Rank : 4
Schlafli Type : {4,20,6}
Number of vertices, edges, etc : 4, 80, 120, 12
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 : {2,20,6}*960c
   4-fold quotients : {4,10,6}*480, {2,20,6}*480b
   5-fold quotients : {4,4,6}*384d
   8-fold quotients : {2,10,6}*240
   10-fold quotients : {4,4,3}*192b, {2,4,6}*192
   12-fold quotients : {4,10,2}*160
   20-fold quotients : {4,2,6}*96, {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   24-fold quotients : {2,10,2}*80
   40-fold quotients : {4,2,3}*48, {2,4,3}*48, {2,2,6}*48
   48-fold quotients : {2,5,2}*40
   60-fold quotients : {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 := (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 := (  1,123)(  2,124)(  3,121)(  4,122)(  5,139)(  6,140)(  7,137)(  8,138)
(  9,135)( 10,136)( 11,133)( 12,134)( 13,131)( 14,132)( 15,129)( 16,130)
( 17,127)( 18,128)( 19,125)( 20,126)( 21,143)( 22,144)( 23,141)( 24,142)
( 25,159)( 26,160)( 27,157)( 28,158)( 29,155)( 30,156)( 31,153)( 32,154)
( 33,151)( 34,152)( 35,149)( 36,150)( 37,147)( 38,148)( 39,145)( 40,146)
( 41,163)( 42,164)( 43,161)( 44,162)( 45,179)( 46,180)( 47,177)( 48,178)
( 49,175)( 50,176)( 51,173)( 52,174)( 53,171)( 54,172)( 55,169)( 56,170)
( 57,167)( 58,168)( 59,165)( 60,166)( 61,183)( 62,184)( 63,181)( 64,182)
( 65,199)( 66,200)( 67,197)( 68,198)( 69,195)( 70,196)( 71,193)( 72,194)
( 73,191)( 74,192)( 75,189)( 76,190)( 77,187)( 78,188)( 79,185)( 80,186)
( 81,203)( 82,204)( 83,201)( 84,202)( 85,219)( 86,220)( 87,217)( 88,218)
( 89,215)( 90,216)( 91,213)( 92,214)( 93,211)( 94,212)( 95,209)( 96,210)
( 97,207)( 98,208)( 99,205)(100,206)(101,223)(102,224)(103,221)(104,222)
(105,239)(106,240)(107,237)(108,238)(109,235)(110,236)(111,233)(112,234)
(113,231)(114,232)(115,229)(116,230)(117,227)(118,228)(119,225)(120,226);;
s2 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 17)( 10, 18)( 11, 20)( 12, 19)
( 15, 16)( 21, 45)( 22, 46)( 23, 48)( 24, 47)( 25, 41)( 26, 42)( 27, 44)
( 28, 43)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)
( 36, 55)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 61, 65)( 62, 66)( 63, 68)
( 64, 67)( 69, 77)( 70, 78)( 71, 80)( 72, 79)( 75, 76)( 81,105)( 82,106)
( 83,108)( 84,107)( 85,101)( 86,102)( 87,104)( 88,103)( 89,117)( 90,118)
( 91,120)( 92,119)( 93,113)( 94,114)( 95,116)( 96,115)( 97,109)( 98,110)
( 99,112)(100,111)(121,125)(122,126)(123,128)(124,127)(129,137)(130,138)
(131,140)(132,139)(135,136)(141,165)(142,166)(143,168)(144,167)(145,161)
(146,162)(147,164)(148,163)(149,177)(150,178)(151,180)(152,179)(153,173)
(154,174)(155,176)(156,175)(157,169)(158,170)(159,172)(160,171)(181,185)
(182,186)(183,188)(184,187)(189,197)(190,198)(191,200)(192,199)(195,196)
(201,225)(202,226)(203,228)(204,227)(205,221)(206,222)(207,224)(208,223)
(209,237)(210,238)(211,240)(212,239)(213,233)(214,234)(215,236)(216,235)
(217,229)(218,230)(219,232)(220,231);;
s3 := (  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)(  8, 26)
(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)( 16, 34)
( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)( 54, 56)
( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)( 67, 87)
( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)( 75, 95)
( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)(110,112)
(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)(126,148)
(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)(134,156)
(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)(166,168)
(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)(185,205)
(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)(193,213)
(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)(222,224)
(226,228)(230,232)(234,236)(238,240);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*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)!(  1,123)(  2,124)(  3,121)(  4,122)(  5,139)(  6,140)(  7,137)
(  8,138)(  9,135)( 10,136)( 11,133)( 12,134)( 13,131)( 14,132)( 15,129)
( 16,130)( 17,127)( 18,128)( 19,125)( 20,126)( 21,143)( 22,144)( 23,141)
( 24,142)( 25,159)( 26,160)( 27,157)( 28,158)( 29,155)( 30,156)( 31,153)
( 32,154)( 33,151)( 34,152)( 35,149)( 36,150)( 37,147)( 38,148)( 39,145)
( 40,146)( 41,163)( 42,164)( 43,161)( 44,162)( 45,179)( 46,180)( 47,177)
( 48,178)( 49,175)( 50,176)( 51,173)( 52,174)( 53,171)( 54,172)( 55,169)
( 56,170)( 57,167)( 58,168)( 59,165)( 60,166)( 61,183)( 62,184)( 63,181)
( 64,182)( 65,199)( 66,200)( 67,197)( 68,198)( 69,195)( 70,196)( 71,193)
( 72,194)( 73,191)( 74,192)( 75,189)( 76,190)( 77,187)( 78,188)( 79,185)
( 80,186)( 81,203)( 82,204)( 83,201)( 84,202)( 85,219)( 86,220)( 87,217)
( 88,218)( 89,215)( 90,216)( 91,213)( 92,214)( 93,211)( 94,212)( 95,209)
( 96,210)( 97,207)( 98,208)( 99,205)(100,206)(101,223)(102,224)(103,221)
(104,222)(105,239)(106,240)(107,237)(108,238)(109,235)(110,236)(111,233)
(112,234)(113,231)(114,232)(115,229)(116,230)(117,227)(118,228)(119,225)
(120,226);
s2 := Sym(240)!(  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 17)( 10, 18)( 11, 20)
( 12, 19)( 15, 16)( 21, 45)( 22, 46)( 23, 48)( 24, 47)( 25, 41)( 26, 42)
( 27, 44)( 28, 43)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)
( 35, 56)( 36, 55)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 61, 65)( 62, 66)
( 63, 68)( 64, 67)( 69, 77)( 70, 78)( 71, 80)( 72, 79)( 75, 76)( 81,105)
( 82,106)( 83,108)( 84,107)( 85,101)( 86,102)( 87,104)( 88,103)( 89,117)
( 90,118)( 91,120)( 92,119)( 93,113)( 94,114)( 95,116)( 96,115)( 97,109)
( 98,110)( 99,112)(100,111)(121,125)(122,126)(123,128)(124,127)(129,137)
(130,138)(131,140)(132,139)(135,136)(141,165)(142,166)(143,168)(144,167)
(145,161)(146,162)(147,164)(148,163)(149,177)(150,178)(151,180)(152,179)
(153,173)(154,174)(155,176)(156,175)(157,169)(158,170)(159,172)(160,171)
(181,185)(182,186)(183,188)(184,187)(189,197)(190,198)(191,200)(192,199)
(195,196)(201,225)(202,226)(203,228)(204,227)(205,221)(206,222)(207,224)
(208,223)(209,237)(210,238)(211,240)(212,239)(213,233)(214,234)(215,236)
(216,235)(217,229)(218,230)(219,232)(220,231);
s3 := Sym(240)!(  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)
(  8, 26)(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)
( 16, 34)( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)
( 54, 56)( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)
( 67, 87)( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)
( 75, 95)( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)
(110,112)(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)
(126,148)(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)
(134,156)(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)
(166,168)(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)
(185,205)(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)
(193,213)(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)
(222,224)(226,228)(230,232)(234,236)(238,240);
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 >;

References : None.
to this polytope