Polytope of Type {4,60}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,60}*1920d
if this polytope has a name.
Group : SmallGroup(1920,239398)
Rank : 3
Schlafli Type : {4,60}
Number of vertices, edges, etc : 16, 480, 240
Order of s0s1s2 : 60
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {4,60}*960b, {4,30}*960b, {4,60}*960c
4-fold quotients : {4,60}*480a, {4,60}*480b, {4,60}*480c, {4,30}*480
5-fold quotients : {4,12}*384d
8-fold quotients : {2,60}*240, {4,30}*240a, {4,15}*240, {4,30}*240b, {4,30}*240c
10-fold quotients : {4,12}*192b, {4,6}*192b, {4,12}*192c
12-fold quotients : {4,20}*160
16-fold quotients : {4,15}*120, {2,30}*120
20-fold quotients : {4,12}*96a, {4,12}*96b, {4,12}*96c, {4,6}*96
24-fold quotients : {2,20}*80, {4,10}*80
32-fold quotients : {2,15}*60
40-fold quotients : {2,12}*48, {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {2,10}*40
60-fold quotients : {4,4}*32
80-fold quotients : {4,3}*24, {2,6}*24
96-fold quotients : {2,5}*20
120-fold quotients : {2,4}*16, {4,2}*16
160-fold quotients : {2,3}*12
240-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,363)(242,364)(243,361)(244,362)(245,367)(246,368)(247,365)(248,366)
(249,371)(250,372)(251,369)(252,370)(253,375)(254,376)(255,373)(256,374)
(257,379)(258,380)(259,377)(260,378)(261,383)(262,384)(263,381)(264,382)
(265,387)(266,388)(267,385)(268,386)(269,391)(270,392)(271,389)(272,390)
(273,395)(274,396)(275,393)(276,394)(277,399)(278,400)(279,397)(280,398)
(281,403)(282,404)(283,401)(284,402)(285,407)(286,408)(287,405)(288,406)
(289,411)(290,412)(291,409)(292,410)(293,415)(294,416)(295,413)(296,414)
(297,419)(298,420)(299,417)(300,418)(301,423)(302,424)(303,421)(304,422)
(305,427)(306,428)(307,425)(308,426)(309,431)(310,432)(311,429)(312,430)
(313,435)(314,436)(315,433)(316,434)(317,439)(318,440)(319,437)(320,438)
(321,443)(322,444)(323,441)(324,442)(325,447)(326,448)(327,445)(328,446)
(329,451)(330,452)(331,449)(332,450)(333,455)(334,456)(335,453)(336,454)
(337,459)(338,460)(339,457)(340,458)(341,463)(342,464)(343,461)(344,462)
(345,467)(346,468)(347,465)(348,466)(349,471)(350,472)(351,469)(352,470)
(353,475)(354,476)(355,473)(356,474)(357,479)(358,480)(359,477)(360,478);;
s1 := ( 1,241)( 2,242)( 3,244)( 4,243)( 5,257)( 6,258)( 7,260)( 8,259)
( 9,253)( 10,254)( 11,256)( 12,255)( 13,249)( 14,250)( 15,252)( 16,251)
( 17,245)( 18,246)( 19,248)( 20,247)( 21,281)( 22,282)( 23,284)( 24,283)
( 25,297)( 26,298)( 27,300)( 28,299)( 29,293)( 30,294)( 31,296)( 32,295)
( 33,289)( 34,290)( 35,292)( 36,291)( 37,285)( 38,286)( 39,288)( 40,287)
( 41,261)( 42,262)( 43,264)( 44,263)( 45,277)( 46,278)( 47,280)( 48,279)
( 49,273)( 50,274)( 51,276)( 52,275)( 53,269)( 54,270)( 55,272)( 56,271)
( 57,265)( 58,266)( 59,268)( 60,267)( 61,301)( 62,302)( 63,304)( 64,303)
( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)( 72,315)
( 73,309)( 74,310)( 75,312)( 76,311)( 77,305)( 78,306)( 79,308)( 80,307)
( 81,341)( 82,342)( 83,344)( 84,343)( 85,357)( 86,358)( 87,360)( 88,359)
( 89,353)( 90,354)( 91,356)( 92,355)( 93,349)( 94,350)( 95,352)( 96,351)
( 97,345)( 98,346)( 99,348)(100,347)(101,321)(102,322)(103,324)(104,323)
(105,337)(106,338)(107,340)(108,339)(109,333)(110,334)(111,336)(112,335)
(113,329)(114,330)(115,332)(116,331)(117,325)(118,326)(119,328)(120,327)
(121,361)(122,362)(123,364)(124,363)(125,377)(126,378)(127,380)(128,379)
(129,373)(130,374)(131,376)(132,375)(133,369)(134,370)(135,372)(136,371)
(137,365)(138,366)(139,368)(140,367)(141,401)(142,402)(143,404)(144,403)
(145,417)(146,418)(147,420)(148,419)(149,413)(150,414)(151,416)(152,415)
(153,409)(154,410)(155,412)(156,411)(157,405)(158,406)(159,408)(160,407)
(161,381)(162,382)(163,384)(164,383)(165,397)(166,398)(167,400)(168,399)
(169,393)(170,394)(171,396)(172,395)(173,389)(174,390)(175,392)(176,391)
(177,385)(178,386)(179,388)(180,387)(181,421)(182,422)(183,424)(184,423)
(185,437)(186,438)(187,440)(188,439)(189,433)(190,434)(191,436)(192,435)
(193,429)(194,430)(195,432)(196,431)(197,425)(198,426)(199,428)(200,427)
(201,461)(202,462)(203,464)(204,463)(205,477)(206,478)(207,480)(208,479)
(209,473)(210,474)(211,476)(212,475)(213,469)(214,470)(215,472)(216,471)
(217,465)(218,466)(219,468)(220,467)(221,441)(222,442)(223,444)(224,443)
(225,457)(226,458)(227,460)(228,459)(229,453)(230,454)(231,456)(232,455)
(233,449)(234,450)(235,452)(236,451)(237,445)(238,446)(239,448)(240,447);;
s2 := ( 1, 25)( 2, 28)( 3, 27)( 4, 26)( 5, 21)( 6, 24)( 7, 23)( 8, 22)
( 9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 33)( 14, 36)( 15, 35)( 16, 34)
( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 41, 45)( 42, 48)( 43, 47)( 44, 46)
( 49, 57)( 50, 60)( 51, 59)( 52, 58)( 54, 56)( 61, 85)( 62, 88)( 63, 87)
( 64, 86)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 97)( 70,100)( 71, 99)
( 72, 98)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 89)( 78, 92)( 79, 91)
( 80, 90)(101,105)(102,108)(103,107)(104,106)(109,117)(110,120)(111,119)
(112,118)(114,116)(121,145)(122,148)(123,147)(124,146)(125,141)(126,144)
(127,143)(128,142)(129,157)(130,160)(131,159)(132,158)(133,153)(134,156)
(135,155)(136,154)(137,149)(138,152)(139,151)(140,150)(161,165)(162,168)
(163,167)(164,166)(169,177)(170,180)(171,179)(172,178)(174,176)(181,205)
(182,208)(183,207)(184,206)(185,201)(186,204)(187,203)(188,202)(189,217)
(190,220)(191,219)(192,218)(193,213)(194,216)(195,215)(196,214)(197,209)
(198,212)(199,211)(200,210)(221,225)(222,228)(223,227)(224,226)(229,237)
(230,240)(231,239)(232,238)(234,236)(241,325)(242,328)(243,327)(244,326)
(245,321)(246,324)(247,323)(248,322)(249,337)(250,340)(251,339)(252,338)
(253,333)(254,336)(255,335)(256,334)(257,329)(258,332)(259,331)(260,330)
(261,305)(262,308)(263,307)(264,306)(265,301)(266,304)(267,303)(268,302)
(269,317)(270,320)(271,319)(272,318)(273,313)(274,316)(275,315)(276,314)
(277,309)(278,312)(279,311)(280,310)(281,345)(282,348)(283,347)(284,346)
(285,341)(286,344)(287,343)(288,342)(289,357)(290,360)(291,359)(292,358)
(293,353)(294,356)(295,355)(296,354)(297,349)(298,352)(299,351)(300,350)
(361,445)(362,448)(363,447)(364,446)(365,441)(366,444)(367,443)(368,442)
(369,457)(370,460)(371,459)(372,458)(373,453)(374,456)(375,455)(376,454)
(377,449)(378,452)(379,451)(380,450)(381,425)(382,428)(383,427)(384,426)
(385,421)(386,424)(387,423)(388,422)(389,437)(390,440)(391,439)(392,438)
(393,433)(394,436)(395,435)(396,434)(397,429)(398,432)(399,431)(400,430)
(401,465)(402,468)(403,467)(404,466)(405,461)(406,464)(407,463)(408,462)
(409,477)(410,480)(411,479)(412,478)(413,473)(414,476)(415,475)(416,474)
(417,469)(418,472)(419,471)(420,470);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*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*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*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(480)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,363)(242,364)(243,361)(244,362)(245,367)(246,368)(247,365)
(248,366)(249,371)(250,372)(251,369)(252,370)(253,375)(254,376)(255,373)
(256,374)(257,379)(258,380)(259,377)(260,378)(261,383)(262,384)(263,381)
(264,382)(265,387)(266,388)(267,385)(268,386)(269,391)(270,392)(271,389)
(272,390)(273,395)(274,396)(275,393)(276,394)(277,399)(278,400)(279,397)
(280,398)(281,403)(282,404)(283,401)(284,402)(285,407)(286,408)(287,405)
(288,406)(289,411)(290,412)(291,409)(292,410)(293,415)(294,416)(295,413)
(296,414)(297,419)(298,420)(299,417)(300,418)(301,423)(302,424)(303,421)
(304,422)(305,427)(306,428)(307,425)(308,426)(309,431)(310,432)(311,429)
(312,430)(313,435)(314,436)(315,433)(316,434)(317,439)(318,440)(319,437)
(320,438)(321,443)(322,444)(323,441)(324,442)(325,447)(326,448)(327,445)
(328,446)(329,451)(330,452)(331,449)(332,450)(333,455)(334,456)(335,453)
(336,454)(337,459)(338,460)(339,457)(340,458)(341,463)(342,464)(343,461)
(344,462)(345,467)(346,468)(347,465)(348,466)(349,471)(350,472)(351,469)
(352,470)(353,475)(354,476)(355,473)(356,474)(357,479)(358,480)(359,477)
(360,478);
s1 := Sym(480)!( 1,241)( 2,242)( 3,244)( 4,243)( 5,257)( 6,258)( 7,260)
( 8,259)( 9,253)( 10,254)( 11,256)( 12,255)( 13,249)( 14,250)( 15,252)
( 16,251)( 17,245)( 18,246)( 19,248)( 20,247)( 21,281)( 22,282)( 23,284)
( 24,283)( 25,297)( 26,298)( 27,300)( 28,299)( 29,293)( 30,294)( 31,296)
( 32,295)( 33,289)( 34,290)( 35,292)( 36,291)( 37,285)( 38,286)( 39,288)
( 40,287)( 41,261)( 42,262)( 43,264)( 44,263)( 45,277)( 46,278)( 47,280)
( 48,279)( 49,273)( 50,274)( 51,276)( 52,275)( 53,269)( 54,270)( 55,272)
( 56,271)( 57,265)( 58,266)( 59,268)( 60,267)( 61,301)( 62,302)( 63,304)
( 64,303)( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)
( 72,315)( 73,309)( 74,310)( 75,312)( 76,311)( 77,305)( 78,306)( 79,308)
( 80,307)( 81,341)( 82,342)( 83,344)( 84,343)( 85,357)( 86,358)( 87,360)
( 88,359)( 89,353)( 90,354)( 91,356)( 92,355)( 93,349)( 94,350)( 95,352)
( 96,351)( 97,345)( 98,346)( 99,348)(100,347)(101,321)(102,322)(103,324)
(104,323)(105,337)(106,338)(107,340)(108,339)(109,333)(110,334)(111,336)
(112,335)(113,329)(114,330)(115,332)(116,331)(117,325)(118,326)(119,328)
(120,327)(121,361)(122,362)(123,364)(124,363)(125,377)(126,378)(127,380)
(128,379)(129,373)(130,374)(131,376)(132,375)(133,369)(134,370)(135,372)
(136,371)(137,365)(138,366)(139,368)(140,367)(141,401)(142,402)(143,404)
(144,403)(145,417)(146,418)(147,420)(148,419)(149,413)(150,414)(151,416)
(152,415)(153,409)(154,410)(155,412)(156,411)(157,405)(158,406)(159,408)
(160,407)(161,381)(162,382)(163,384)(164,383)(165,397)(166,398)(167,400)
(168,399)(169,393)(170,394)(171,396)(172,395)(173,389)(174,390)(175,392)
(176,391)(177,385)(178,386)(179,388)(180,387)(181,421)(182,422)(183,424)
(184,423)(185,437)(186,438)(187,440)(188,439)(189,433)(190,434)(191,436)
(192,435)(193,429)(194,430)(195,432)(196,431)(197,425)(198,426)(199,428)
(200,427)(201,461)(202,462)(203,464)(204,463)(205,477)(206,478)(207,480)
(208,479)(209,473)(210,474)(211,476)(212,475)(213,469)(214,470)(215,472)
(216,471)(217,465)(218,466)(219,468)(220,467)(221,441)(222,442)(223,444)
(224,443)(225,457)(226,458)(227,460)(228,459)(229,453)(230,454)(231,456)
(232,455)(233,449)(234,450)(235,452)(236,451)(237,445)(238,446)(239,448)
(240,447);
s2 := Sym(480)!( 1, 25)( 2, 28)( 3, 27)( 4, 26)( 5, 21)( 6, 24)( 7, 23)
( 8, 22)( 9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 33)( 14, 36)( 15, 35)
( 16, 34)( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 41, 45)( 42, 48)( 43, 47)
( 44, 46)( 49, 57)( 50, 60)( 51, 59)( 52, 58)( 54, 56)( 61, 85)( 62, 88)
( 63, 87)( 64, 86)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 97)( 70,100)
( 71, 99)( 72, 98)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 89)( 78, 92)
( 79, 91)( 80, 90)(101,105)(102,108)(103,107)(104,106)(109,117)(110,120)
(111,119)(112,118)(114,116)(121,145)(122,148)(123,147)(124,146)(125,141)
(126,144)(127,143)(128,142)(129,157)(130,160)(131,159)(132,158)(133,153)
(134,156)(135,155)(136,154)(137,149)(138,152)(139,151)(140,150)(161,165)
(162,168)(163,167)(164,166)(169,177)(170,180)(171,179)(172,178)(174,176)
(181,205)(182,208)(183,207)(184,206)(185,201)(186,204)(187,203)(188,202)
(189,217)(190,220)(191,219)(192,218)(193,213)(194,216)(195,215)(196,214)
(197,209)(198,212)(199,211)(200,210)(221,225)(222,228)(223,227)(224,226)
(229,237)(230,240)(231,239)(232,238)(234,236)(241,325)(242,328)(243,327)
(244,326)(245,321)(246,324)(247,323)(248,322)(249,337)(250,340)(251,339)
(252,338)(253,333)(254,336)(255,335)(256,334)(257,329)(258,332)(259,331)
(260,330)(261,305)(262,308)(263,307)(264,306)(265,301)(266,304)(267,303)
(268,302)(269,317)(270,320)(271,319)(272,318)(273,313)(274,316)(275,315)
(276,314)(277,309)(278,312)(279,311)(280,310)(281,345)(282,348)(283,347)
(284,346)(285,341)(286,344)(287,343)(288,342)(289,357)(290,360)(291,359)
(292,358)(293,353)(294,356)(295,355)(296,354)(297,349)(298,352)(299,351)
(300,350)(361,445)(362,448)(363,447)(364,446)(365,441)(366,444)(367,443)
(368,442)(369,457)(370,460)(371,459)(372,458)(373,453)(374,456)(375,455)
(376,454)(377,449)(378,452)(379,451)(380,450)(381,425)(382,428)(383,427)
(384,426)(385,421)(386,424)(387,423)(388,422)(389,437)(390,440)(391,439)
(392,438)(393,433)(394,436)(395,435)(396,434)(397,429)(398,432)(399,431)
(400,430)(401,465)(402,468)(403,467)(404,466)(405,461)(406,464)(407,463)
(408,462)(409,477)(410,480)(411,479)(412,478)(413,473)(414,476)(415,475)
(416,474)(417,469)(418,472)(419,471)(420,470);
poly := sub<Sym(480)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*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*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*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