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Polytope of Type {20,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,6,4}*1920b
if this polytope has a name.
Group : SmallGroup(1920,240151)
Rank : 4
Schlafli Type : {20,6,4}
Number of vertices, edges, etc : 40, 120, 24, 4
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 : {20,6,2}*960c
   4-fold quotients : {10,6,4}*480a, {20,6,2}*480b
   5-fold quotients : {4,6,4}*384b
   8-fold quotients : {10,6,2}*240
   10-fold quotients : {4,6,4}*192c, {4,6,2}*192
   12-fold quotients : {10,2,4}*160
   20-fold quotients : {2,6,4}*96a, {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
   24-fold quotients : {5,2,4}*80, {10,2,2}*80
   40-fold quotients : {4,3,2}*48, {2,6,2}*48
   48-fold quotients : {5,2,2}*40
   60-fold quotients : {2,2,4}*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 := (  1,  3)(  2,  4)(  5, 19)(  6, 20)(  7, 17)(  8, 18)(  9, 15)( 10, 16)
( 11, 13)( 12, 14)( 21, 23)( 22, 24)( 25, 39)( 26, 40)( 27, 37)( 28, 38)
( 29, 35)( 30, 36)( 31, 33)( 32, 34)( 41, 43)( 42, 44)( 45, 59)( 46, 60)
( 47, 57)( 48, 58)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 61, 63)( 62, 64)
( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)( 72, 74)
( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89, 95)( 90, 96)
( 91, 93)( 92, 94)(101,103)(102,104)(105,119)(106,120)(107,117)(108,118)
(109,115)(110,116)(111,113)(112,114)(121,123)(122,124)(125,139)(126,140)
(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(141,143)(142,144)
(145,159)(146,160)(147,157)(148,158)(149,155)(150,156)(151,153)(152,154)
(161,163)(162,164)(165,179)(166,180)(167,177)(168,178)(169,175)(170,176)
(171,173)(172,174)(181,183)(182,184)(185,199)(186,200)(187,197)(188,198)
(189,195)(190,196)(191,193)(192,194)(201,203)(202,204)(205,219)(206,220)
(207,217)(208,218)(209,215)(210,216)(211,213)(212,214)(221,223)(222,224)
(225,239)(226,240)(227,237)(228,238)(229,235)(230,236)(231,233)(232,234);;
s1 := (  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);;
s2 := (  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)(121,201)(122,204)(123,203)(124,202)(125,205)(126,208)
(127,207)(128,206)(129,209)(130,212)(131,211)(132,210)(133,213)(134,216)
(135,215)(136,214)(137,217)(138,220)(139,219)(140,218)(141,181)(142,184)
(143,183)(144,182)(145,185)(146,188)(147,187)(148,186)(149,189)(150,192)
(151,191)(152,190)(153,193)(154,196)(155,195)(156,194)(157,197)(158,200)
(159,199)(160,198)(161,221)(162,224)(163,223)(164,222)(165,225)(166,228)
(167,227)(168,226)(169,229)(170,232)(171,231)(172,230)(173,233)(174,236)
(175,235)(176,234)(177,237)(178,240)(179,239)(180,238);;
s3 := (  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)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)
( 81,201)( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)( 88,208)
( 89,209)( 90,210)( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)( 96,216)
( 97,217)( 98,218)( 99,219)(100,220)(101,221)(102,222)(103,223)(104,224)
(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)
(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(240)!(  1,  3)(  2,  4)(  5, 19)(  6, 20)(  7, 17)(  8, 18)(  9, 15)
( 10, 16)( 11, 13)( 12, 14)( 21, 23)( 22, 24)( 25, 39)( 26, 40)( 27, 37)
( 28, 38)( 29, 35)( 30, 36)( 31, 33)( 32, 34)( 41, 43)( 42, 44)( 45, 59)
( 46, 60)( 47, 57)( 48, 58)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 61, 63)
( 62, 64)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)
( 72, 74)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89, 95)
( 90, 96)( 91, 93)( 92, 94)(101,103)(102,104)(105,119)(106,120)(107,117)
(108,118)(109,115)(110,116)(111,113)(112,114)(121,123)(122,124)(125,139)
(126,140)(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(141,143)
(142,144)(145,159)(146,160)(147,157)(148,158)(149,155)(150,156)(151,153)
(152,154)(161,163)(162,164)(165,179)(166,180)(167,177)(168,178)(169,175)
(170,176)(171,173)(172,174)(181,183)(182,184)(185,199)(186,200)(187,197)
(188,198)(189,195)(190,196)(191,193)(192,194)(201,203)(202,204)(205,219)
(206,220)(207,217)(208,218)(209,215)(210,216)(211,213)(212,214)(221,223)
(222,224)(225,239)(226,240)(227,237)(228,238)(229,235)(230,236)(231,233)
(232,234);
s1 := 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);
s2 := 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)(121,201)(122,204)(123,203)(124,202)(125,205)
(126,208)(127,207)(128,206)(129,209)(130,212)(131,211)(132,210)(133,213)
(134,216)(135,215)(136,214)(137,217)(138,220)(139,219)(140,218)(141,181)
(142,184)(143,183)(144,182)(145,185)(146,188)(147,187)(148,186)(149,189)
(150,192)(151,191)(152,190)(153,193)(154,196)(155,195)(156,194)(157,197)
(158,200)(159,199)(160,198)(161,221)(162,224)(163,223)(164,222)(165,225)
(166,228)(167,227)(168,226)(169,229)(170,232)(171,231)(172,230)(173,233)
(174,236)(175,235)(176,234)(177,237)(178,240)(179,239)(180,238);
s3 := 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)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)
( 80,200)( 81,201)( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)
( 88,208)( 89,209)( 90,210)( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)
( 96,216)( 97,217)( 98,218)( 99,219)(100,220)(101,221)(102,222)(103,223)
(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)
(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)
(120,240);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 >; 
 
References : None.
to this polytope