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# Polytope of Type {4,60}

Atlas Canonical Name : {4,60}*960b
if this polytope has a name.
Group : SmallGroup(960,11090)
Rank : 3
Schlafli Type : {4,60}
Number of vertices, edges, etc : 8, 240, 120
Order of s0s1s2 : 60
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,60,2} of size 1920
Vertex Figure Of :
{2,4,60} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,60}*480b, {4,60}*480c, {4,30}*480
4-fold quotients : {2,60}*240, {4,15}*240, {4,30}*240b, {4,30}*240c
5-fold quotients : {4,12}*192b
8-fold quotients : {4,15}*120, {2,30}*120
10-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
12-fold quotients : {2,20}*80
16-fold quotients : {2,15}*60
20-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {2,10}*40
40-fold quotients : {4,3}*24, {2,6}*24
48-fold quotients : {2,5}*20
60-fold quotients : {2,4}*16
80-fold quotients : {2,3}*12
120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,60}*1920d, {8,60}*1920e, {8,60}*1920f, {4,120}*1920c, {4,120}*1920d
Permutation Representation (GAP) :
```s0 := (  1,243)(  2,244)(  3,241)(  4,242)(  5,247)(  6,248)(  7,245)(  8,246)
(  9,251)( 10,252)( 11,249)( 12,250)( 13,255)( 14,256)( 15,253)( 16,254)
( 17,259)( 18,260)( 19,257)( 20,258)( 21,263)( 22,264)( 23,261)( 24,262)
( 25,267)( 26,268)( 27,265)( 28,266)( 29,271)( 30,272)( 31,269)( 32,270)
( 33,275)( 34,276)( 35,273)( 36,274)( 37,279)( 38,280)( 39,277)( 40,278)
( 41,283)( 42,284)( 43,281)( 44,282)( 45,287)( 46,288)( 47,285)( 48,286)
( 49,291)( 50,292)( 51,289)( 52,290)( 53,295)( 54,296)( 55,293)( 56,294)
( 57,299)( 58,300)( 59,297)( 60,298)( 61,303)( 62,304)( 63,301)( 64,302)
( 65,307)( 66,308)( 67,305)( 68,306)( 69,311)( 70,312)( 71,309)( 72,310)
( 73,315)( 74,316)( 75,313)( 76,314)( 77,319)( 78,320)( 79,317)( 80,318)
( 81,323)( 82,324)( 83,321)( 84,322)( 85,327)( 86,328)( 87,325)( 88,326)
( 89,331)( 90,332)( 91,329)( 92,330)( 93,335)( 94,336)( 95,333)( 96,334)
( 97,339)( 98,340)( 99,337)(100,338)(101,343)(102,344)(103,341)(104,342)
(105,347)(106,348)(107,345)(108,346)(109,351)(110,352)(111,349)(112,350)
(113,355)(114,356)(115,353)(116,354)(117,359)(118,360)(119,357)(120,358)
(121,363)(122,364)(123,361)(124,362)(125,367)(126,368)(127,365)(128,366)
(129,371)(130,372)(131,369)(132,370)(133,375)(134,376)(135,373)(136,374)
(137,379)(138,380)(139,377)(140,378)(141,383)(142,384)(143,381)(144,382)
(145,387)(146,388)(147,385)(148,386)(149,391)(150,392)(151,389)(152,390)
(153,395)(154,396)(155,393)(156,394)(157,399)(158,400)(159,397)(160,398)
(161,403)(162,404)(163,401)(164,402)(165,407)(166,408)(167,405)(168,406)
(169,411)(170,412)(171,409)(172,410)(173,415)(174,416)(175,413)(176,414)
(177,419)(178,420)(179,417)(180,418)(181,423)(182,424)(183,421)(184,422)
(185,427)(186,428)(187,425)(188,426)(189,431)(190,432)(191,429)(192,430)
(193,435)(194,436)(195,433)(196,434)(197,439)(198,440)(199,437)(200,438)
(201,443)(202,444)(203,441)(204,442)(205,447)(206,448)(207,445)(208,446)
(209,451)(210,452)(211,449)(212,450)(213,455)(214,456)(215,453)(216,454)
(217,459)(218,460)(219,457)(220,458)(221,463)(222,464)(223,461)(224,462)
(225,467)(226,468)(227,465)(228,466)(229,471)(230,472)(231,469)(232,470)
(233,475)(234,476)(235,473)(236,474)(237,479)(238,480)(239,477)(240,478);;
s1 := (  2,  3)(  5, 17)(  6, 19)(  7, 18)(  8, 20)(  9, 13)( 10, 15)( 11, 14)
( 12, 16)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 57)( 26, 59)( 27, 58)
( 28, 60)( 29, 53)( 30, 55)( 31, 54)( 32, 56)( 33, 49)( 34, 51)( 35, 50)
( 36, 52)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 62, 63)( 65, 77)( 66, 79)
( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,117)( 86,119)( 87,118)( 88,120)( 89,113)( 90,115)
( 91,114)( 92,116)( 93,109)( 94,111)( 95,110)( 96,112)( 97,105)( 98,107)
( 99,106)(100,108)(121,181)(122,183)(123,182)(124,184)(125,197)(126,199)
(127,198)(128,200)(129,193)(130,195)(131,194)(132,196)(133,189)(134,191)
(135,190)(136,192)(137,185)(138,187)(139,186)(140,188)(141,221)(142,223)
(143,222)(144,224)(145,237)(146,239)(147,238)(148,240)(149,233)(150,235)
(151,234)(152,236)(153,229)(154,231)(155,230)(156,232)(157,225)(158,227)
(159,226)(160,228)(161,201)(162,203)(163,202)(164,204)(165,217)(166,219)
(167,218)(168,220)(169,213)(170,215)(171,214)(172,216)(173,209)(174,211)
(175,210)(176,212)(177,205)(178,207)(179,206)(180,208)(242,243)(245,257)
(246,259)(247,258)(248,260)(249,253)(250,255)(251,254)(252,256)(261,281)
(262,283)(263,282)(264,284)(265,297)(266,299)(267,298)(268,300)(269,293)
(270,295)(271,294)(272,296)(273,289)(274,291)(275,290)(276,292)(277,285)
(278,287)(279,286)(280,288)(302,303)(305,317)(306,319)(307,318)(308,320)
(309,313)(310,315)(311,314)(312,316)(321,341)(322,343)(323,342)(324,344)
(325,357)(326,359)(327,358)(328,360)(329,353)(330,355)(331,354)(332,356)
(333,349)(334,351)(335,350)(336,352)(337,345)(338,347)(339,346)(340,348)
(361,421)(362,423)(363,422)(364,424)(365,437)(366,439)(367,438)(368,440)
(369,433)(370,435)(371,434)(372,436)(373,429)(374,431)(375,430)(376,432)
(377,425)(378,427)(379,426)(380,428)(381,461)(382,463)(383,462)(384,464)
(385,477)(386,479)(387,478)(388,480)(389,473)(390,475)(391,474)(392,476)
(393,469)(394,471)(395,470)(396,472)(397,465)(398,467)(399,466)(400,468)
(401,441)(402,443)(403,442)(404,444)(405,457)(406,459)(407,458)(408,460)
(409,453)(410,455)(411,454)(412,456)(413,449)(414,451)(415,450)(416,452)
(417,445)(418,447)(419,446)(420,448);;
s2 := (  1,165)(  2,168)(  3,167)(  4,166)(  5,161)(  6,164)(  7,163)(  8,162)
(  9,177)( 10,180)( 11,179)( 12,178)( 13,173)( 14,176)( 15,175)( 16,174)
( 17,169)( 18,172)( 19,171)( 20,170)( 21,145)( 22,148)( 23,147)( 24,146)
( 25,141)( 26,144)( 27,143)( 28,142)( 29,157)( 30,160)( 31,159)( 32,158)
( 33,153)( 34,156)( 35,155)( 36,154)( 37,149)( 38,152)( 39,151)( 40,150)
( 41,125)( 42,128)( 43,127)( 44,126)( 45,121)( 46,124)( 47,123)( 48,122)
( 49,137)( 50,140)( 51,139)( 52,138)( 53,133)( 54,136)( 55,135)( 56,134)
( 57,129)( 58,132)( 59,131)( 60,130)( 61,225)( 62,228)( 63,227)( 64,226)
( 65,221)( 66,224)( 67,223)( 68,222)( 69,237)( 70,240)( 71,239)( 72,238)
( 73,233)( 74,236)( 75,235)( 76,234)( 77,229)( 78,232)( 79,231)( 80,230)
( 81,205)( 82,208)( 83,207)( 84,206)( 85,201)( 86,204)( 87,203)( 88,202)
( 89,217)( 90,220)( 91,219)( 92,218)( 93,213)( 94,216)( 95,215)( 96,214)
( 97,209)( 98,212)( 99,211)(100,210)(101,185)(102,188)(103,187)(104,186)
(105,181)(106,184)(107,183)(108,182)(109,197)(110,200)(111,199)(112,198)
(113,193)(114,196)(115,195)(116,194)(117,189)(118,192)(119,191)(120,190)
(241,405)(242,408)(243,407)(244,406)(245,401)(246,404)(247,403)(248,402)
(249,417)(250,420)(251,419)(252,418)(253,413)(254,416)(255,415)(256,414)
(257,409)(258,412)(259,411)(260,410)(261,385)(262,388)(263,387)(264,386)
(265,381)(266,384)(267,383)(268,382)(269,397)(270,400)(271,399)(272,398)
(273,393)(274,396)(275,395)(276,394)(277,389)(278,392)(279,391)(280,390)
(281,365)(282,368)(283,367)(284,366)(285,361)(286,364)(287,363)(288,362)
(289,377)(290,380)(291,379)(292,378)(293,373)(294,376)(295,375)(296,374)
(297,369)(298,372)(299,371)(300,370)(301,465)(302,468)(303,467)(304,466)
(305,461)(306,464)(307,463)(308,462)(309,477)(310,480)(311,479)(312,478)
(313,473)(314,476)(315,475)(316,474)(317,469)(318,472)(319,471)(320,470)
(321,445)(322,448)(323,447)(324,446)(325,441)(326,444)(327,443)(328,442)
(329,457)(330,460)(331,459)(332,458)(333,453)(334,456)(335,455)(336,454)
(337,449)(338,452)(339,451)(340,450)(341,425)(342,428)(343,427)(344,426)
(345,421)(346,424)(347,423)(348,422)(349,437)(350,440)(351,439)(352,438)
(353,433)(354,436)(355,435)(356,434)(357,429)(358,432)(359,431)(360,430);;
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*s2*s1*s0*s1*s2*s1*s2*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)!(  1,243)(  2,244)(  3,241)(  4,242)(  5,247)(  6,248)(  7,245)
(  8,246)(  9,251)( 10,252)( 11,249)( 12,250)( 13,255)( 14,256)( 15,253)
( 16,254)( 17,259)( 18,260)( 19,257)( 20,258)( 21,263)( 22,264)( 23,261)
( 24,262)( 25,267)( 26,268)( 27,265)( 28,266)( 29,271)( 30,272)( 31,269)
( 32,270)( 33,275)( 34,276)( 35,273)( 36,274)( 37,279)( 38,280)( 39,277)
( 40,278)( 41,283)( 42,284)( 43,281)( 44,282)( 45,287)( 46,288)( 47,285)
( 48,286)( 49,291)( 50,292)( 51,289)( 52,290)( 53,295)( 54,296)( 55,293)
( 56,294)( 57,299)( 58,300)( 59,297)( 60,298)( 61,303)( 62,304)( 63,301)
( 64,302)( 65,307)( 66,308)( 67,305)( 68,306)( 69,311)( 70,312)( 71,309)
( 72,310)( 73,315)( 74,316)( 75,313)( 76,314)( 77,319)( 78,320)( 79,317)
( 80,318)( 81,323)( 82,324)( 83,321)( 84,322)( 85,327)( 86,328)( 87,325)
( 88,326)( 89,331)( 90,332)( 91,329)( 92,330)( 93,335)( 94,336)( 95,333)
( 96,334)( 97,339)( 98,340)( 99,337)(100,338)(101,343)(102,344)(103,341)
(104,342)(105,347)(106,348)(107,345)(108,346)(109,351)(110,352)(111,349)
(112,350)(113,355)(114,356)(115,353)(116,354)(117,359)(118,360)(119,357)
(120,358)(121,363)(122,364)(123,361)(124,362)(125,367)(126,368)(127,365)
(128,366)(129,371)(130,372)(131,369)(132,370)(133,375)(134,376)(135,373)
(136,374)(137,379)(138,380)(139,377)(140,378)(141,383)(142,384)(143,381)
(144,382)(145,387)(146,388)(147,385)(148,386)(149,391)(150,392)(151,389)
(152,390)(153,395)(154,396)(155,393)(156,394)(157,399)(158,400)(159,397)
(160,398)(161,403)(162,404)(163,401)(164,402)(165,407)(166,408)(167,405)
(168,406)(169,411)(170,412)(171,409)(172,410)(173,415)(174,416)(175,413)
(176,414)(177,419)(178,420)(179,417)(180,418)(181,423)(182,424)(183,421)
(184,422)(185,427)(186,428)(187,425)(188,426)(189,431)(190,432)(191,429)
(192,430)(193,435)(194,436)(195,433)(196,434)(197,439)(198,440)(199,437)
(200,438)(201,443)(202,444)(203,441)(204,442)(205,447)(206,448)(207,445)
(208,446)(209,451)(210,452)(211,449)(212,450)(213,455)(214,456)(215,453)
(216,454)(217,459)(218,460)(219,457)(220,458)(221,463)(222,464)(223,461)
(224,462)(225,467)(226,468)(227,465)(228,466)(229,471)(230,472)(231,469)
(232,470)(233,475)(234,476)(235,473)(236,474)(237,479)(238,480)(239,477)
(240,478);
s1 := Sym(480)!(  2,  3)(  5, 17)(  6, 19)(  7, 18)(  8, 20)(  9, 13)( 10, 15)
( 11, 14)( 12, 16)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 57)( 26, 59)
( 27, 58)( 28, 60)( 29, 53)( 30, 55)( 31, 54)( 32, 56)( 33, 49)( 34, 51)
( 35, 50)( 36, 52)( 37, 45)( 38, 47)( 39, 46)( 40, 48)( 62, 63)( 65, 77)
( 66, 79)( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,117)( 86,119)( 87,118)( 88,120)( 89,113)
( 90,115)( 91,114)( 92,116)( 93,109)( 94,111)( 95,110)( 96,112)( 97,105)
( 98,107)( 99,106)(100,108)(121,181)(122,183)(123,182)(124,184)(125,197)
(126,199)(127,198)(128,200)(129,193)(130,195)(131,194)(132,196)(133,189)
(134,191)(135,190)(136,192)(137,185)(138,187)(139,186)(140,188)(141,221)
(142,223)(143,222)(144,224)(145,237)(146,239)(147,238)(148,240)(149,233)
(150,235)(151,234)(152,236)(153,229)(154,231)(155,230)(156,232)(157,225)
(158,227)(159,226)(160,228)(161,201)(162,203)(163,202)(164,204)(165,217)
(166,219)(167,218)(168,220)(169,213)(170,215)(171,214)(172,216)(173,209)
(174,211)(175,210)(176,212)(177,205)(178,207)(179,206)(180,208)(242,243)
(245,257)(246,259)(247,258)(248,260)(249,253)(250,255)(251,254)(252,256)
(261,281)(262,283)(263,282)(264,284)(265,297)(266,299)(267,298)(268,300)
(269,293)(270,295)(271,294)(272,296)(273,289)(274,291)(275,290)(276,292)
(277,285)(278,287)(279,286)(280,288)(302,303)(305,317)(306,319)(307,318)
(308,320)(309,313)(310,315)(311,314)(312,316)(321,341)(322,343)(323,342)
(324,344)(325,357)(326,359)(327,358)(328,360)(329,353)(330,355)(331,354)
(332,356)(333,349)(334,351)(335,350)(336,352)(337,345)(338,347)(339,346)
(340,348)(361,421)(362,423)(363,422)(364,424)(365,437)(366,439)(367,438)
(368,440)(369,433)(370,435)(371,434)(372,436)(373,429)(374,431)(375,430)
(376,432)(377,425)(378,427)(379,426)(380,428)(381,461)(382,463)(383,462)
(384,464)(385,477)(386,479)(387,478)(388,480)(389,473)(390,475)(391,474)
(392,476)(393,469)(394,471)(395,470)(396,472)(397,465)(398,467)(399,466)
(400,468)(401,441)(402,443)(403,442)(404,444)(405,457)(406,459)(407,458)
(408,460)(409,453)(410,455)(411,454)(412,456)(413,449)(414,451)(415,450)
(416,452)(417,445)(418,447)(419,446)(420,448);
s2 := Sym(480)!(  1,165)(  2,168)(  3,167)(  4,166)(  5,161)(  6,164)(  7,163)
(  8,162)(  9,177)( 10,180)( 11,179)( 12,178)( 13,173)( 14,176)( 15,175)
( 16,174)( 17,169)( 18,172)( 19,171)( 20,170)( 21,145)( 22,148)( 23,147)
( 24,146)( 25,141)( 26,144)( 27,143)( 28,142)( 29,157)( 30,160)( 31,159)
( 32,158)( 33,153)( 34,156)( 35,155)( 36,154)( 37,149)( 38,152)( 39,151)
( 40,150)( 41,125)( 42,128)( 43,127)( 44,126)( 45,121)( 46,124)( 47,123)
( 48,122)( 49,137)( 50,140)( 51,139)( 52,138)( 53,133)( 54,136)( 55,135)
( 56,134)( 57,129)( 58,132)( 59,131)( 60,130)( 61,225)( 62,228)( 63,227)
( 64,226)( 65,221)( 66,224)( 67,223)( 68,222)( 69,237)( 70,240)( 71,239)
( 72,238)( 73,233)( 74,236)( 75,235)( 76,234)( 77,229)( 78,232)( 79,231)
( 80,230)( 81,205)( 82,208)( 83,207)( 84,206)( 85,201)( 86,204)( 87,203)
( 88,202)( 89,217)( 90,220)( 91,219)( 92,218)( 93,213)( 94,216)( 95,215)
( 96,214)( 97,209)( 98,212)( 99,211)(100,210)(101,185)(102,188)(103,187)
(104,186)(105,181)(106,184)(107,183)(108,182)(109,197)(110,200)(111,199)
(112,198)(113,193)(114,196)(115,195)(116,194)(117,189)(118,192)(119,191)
(120,190)(241,405)(242,408)(243,407)(244,406)(245,401)(246,404)(247,403)
(248,402)(249,417)(250,420)(251,419)(252,418)(253,413)(254,416)(255,415)
(256,414)(257,409)(258,412)(259,411)(260,410)(261,385)(262,388)(263,387)
(264,386)(265,381)(266,384)(267,383)(268,382)(269,397)(270,400)(271,399)
(272,398)(273,393)(274,396)(275,395)(276,394)(277,389)(278,392)(279,391)
(280,390)(281,365)(282,368)(283,367)(284,366)(285,361)(286,364)(287,363)
(288,362)(289,377)(290,380)(291,379)(292,378)(293,373)(294,376)(295,375)
(296,374)(297,369)(298,372)(299,371)(300,370)(301,465)(302,468)(303,467)
(304,466)(305,461)(306,464)(307,463)(308,462)(309,477)(310,480)(311,479)
(312,478)(313,473)(314,476)(315,475)(316,474)(317,469)(318,472)(319,471)
(320,470)(321,445)(322,448)(323,447)(324,446)(325,441)(326,444)(327,443)
(328,442)(329,457)(330,460)(331,459)(332,458)(333,453)(334,456)(335,455)
(336,454)(337,449)(338,452)(339,451)(340,450)(341,425)(342,428)(343,427)
(344,426)(345,421)(346,424)(347,423)(348,422)(349,437)(350,440)(351,439)
(352,438)(353,433)(354,436)(355,435)(356,434)(357,429)(358,432)(359,431)
(360,430);
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*s2*s1*s0*s1*s2*s1*s2*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