# Polytope of Type {4,120}

Atlas Canonical Name : {4,120}*960d
if this polytope has a name.
Group : SmallGroup(960,6316)
Rank : 3
Schlafli Type : {4,120}
Number of vertices, edges, etc : 4, 240, 120
Order of s0s1s2 : 120
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,120,2} of size 1920
Vertex Figure Of :
{2,4,120} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,60}*480b
4-fold quotients : {4,30}*240b
5-fold quotients : {4,24}*192d
8-fold quotients : {4,15}*120
10-fold quotients : {4,12}*96b
20-fold quotients : {4,6}*48c
40-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,120}*1920c
Permutation Representation (GAP) :
```s0 := (  1, 63)(  2, 64)(  3, 61)(  4, 62)(  5, 67)(  6, 68)(  7, 65)(  8, 66)
(  9, 71)( 10, 72)( 11, 69)( 12, 70)( 13, 75)( 14, 76)( 15, 73)( 16, 74)
( 17, 79)( 18, 80)( 19, 77)( 20, 78)( 21, 83)( 22, 84)( 23, 81)( 24, 82)
( 25, 87)( 26, 88)( 27, 85)( 28, 86)( 29, 91)( 30, 92)( 31, 89)( 32, 90)
( 33, 95)( 34, 96)( 35, 93)( 36, 94)( 37, 99)( 38,100)( 39, 97)( 40, 98)
( 41,103)( 42,104)( 43,101)( 44,102)( 45,107)( 46,108)( 47,105)( 48,106)
( 49,111)( 50,112)( 51,109)( 52,110)( 53,115)( 54,116)( 55,113)( 56,114)
( 57,119)( 58,120)( 59,117)( 60,118)(121,183)(122,184)(123,181)(124,182)
(125,187)(126,188)(127,185)(128,186)(129,191)(130,192)(131,189)(132,190)
(133,195)(134,196)(135,193)(136,194)(137,199)(138,200)(139,197)(140,198)
(141,203)(142,204)(143,201)(144,202)(145,207)(146,208)(147,205)(148,206)
(149,211)(150,212)(151,209)(152,210)(153,215)(154,216)(155,213)(156,214)
(157,219)(158,220)(159,217)(160,218)(161,223)(162,224)(163,221)(164,222)
(165,227)(166,228)(167,225)(168,226)(169,231)(170,232)(171,229)(172,230)
(173,235)(174,236)(175,233)(176,234)(177,239)(178,240)(179,237)(180,238)
(241,303)(242,304)(243,301)(244,302)(245,307)(246,308)(247,305)(248,306)
(249,311)(250,312)(251,309)(252,310)(253,315)(254,316)(255,313)(256,314)
(257,319)(258,320)(259,317)(260,318)(261,323)(262,324)(263,321)(264,322)
(265,327)(266,328)(267,325)(268,326)(269,331)(270,332)(271,329)(272,330)
(273,335)(274,336)(275,333)(276,334)(277,339)(278,340)(279,337)(280,338)
(281,343)(282,344)(283,341)(284,342)(285,347)(286,348)(287,345)(288,346)
(289,351)(290,352)(291,349)(292,350)(293,355)(294,356)(295,353)(296,354)
(297,359)(298,360)(299,357)(300,358)(361,423)(362,424)(363,421)(364,422)
(365,427)(366,428)(367,425)(368,426)(369,431)(370,432)(371,429)(372,430)
(373,435)(374,436)(375,433)(376,434)(377,439)(378,440)(379,437)(380,438)
(381,443)(382,444)(383,441)(384,442)(385,447)(386,448)(387,445)(388,446)
(389,451)(390,452)(391,449)(392,450)(393,455)(394,456)(395,453)(396,454)
(397,459)(398,460)(399,457)(400,458)(401,463)(402,464)(403,461)(404,462)
(405,467)(406,468)(407,465)(408,466)(409,471)(410,472)(411,469)(412,470)
(413,475)(414,476)(415,473)(416,474)(417,479)(418,480)(419,477)(420,478);;
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)(241,361)(242,363)
(243,362)(244,364)(245,377)(246,379)(247,378)(248,380)(249,373)(250,375)
(251,374)(252,376)(253,369)(254,371)(255,370)(256,372)(257,365)(258,367)
(259,366)(260,368)(261,401)(262,403)(263,402)(264,404)(265,417)(266,419)
(267,418)(268,420)(269,413)(270,415)(271,414)(272,416)(273,409)(274,411)
(275,410)(276,412)(277,405)(278,407)(279,406)(280,408)(281,381)(282,383)
(283,382)(284,384)(285,397)(286,399)(287,398)(288,400)(289,393)(290,395)
(291,394)(292,396)(293,389)(294,391)(295,390)(296,392)(297,385)(298,387)
(299,386)(300,388)(301,421)(302,423)(303,422)(304,424)(305,437)(306,439)
(307,438)(308,440)(309,433)(310,435)(311,434)(312,436)(313,429)(314,431)
(315,430)(316,432)(317,425)(318,427)(319,426)(320,428)(321,461)(322,463)
(323,462)(324,464)(325,477)(326,479)(327,478)(328,480)(329,473)(330,475)
(331,474)(332,476)(333,469)(334,471)(335,470)(336,472)(337,465)(338,467)
(339,466)(340,468)(341,441)(342,443)(343,442)(344,444)(345,457)(346,459)
(347,458)(348,460)(349,453)(350,455)(351,454)(352,456)(353,449)(354,451)
(355,450)(356,452)(357,445)(358,447)(359,446)(360,448);;
s2 := (  1,285)(  2,288)(  3,287)(  4,286)(  5,281)(  6,284)(  7,283)(  8,282)
(  9,297)( 10,300)( 11,299)( 12,298)( 13,293)( 14,296)( 15,295)( 16,294)
( 17,289)( 18,292)( 19,291)( 20,290)( 21,265)( 22,268)( 23,267)( 24,266)
( 25,261)( 26,264)( 27,263)( 28,262)( 29,277)( 30,280)( 31,279)( 32,278)
( 33,273)( 34,276)( 35,275)( 36,274)( 37,269)( 38,272)( 39,271)( 40,270)
( 41,245)( 42,248)( 43,247)( 44,246)( 45,241)( 46,244)( 47,243)( 48,242)
( 49,257)( 50,260)( 51,259)( 52,258)( 53,253)( 54,256)( 55,255)( 56,254)
( 57,249)( 58,252)( 59,251)( 60,250)( 61,345)( 62,348)( 63,347)( 64,346)
( 65,341)( 66,344)( 67,343)( 68,342)( 69,357)( 70,360)( 71,359)( 72,358)
( 73,353)( 74,356)( 75,355)( 76,354)( 77,349)( 78,352)( 79,351)( 80,350)
( 81,325)( 82,328)( 83,327)( 84,326)( 85,321)( 86,324)( 87,323)( 88,322)
( 89,337)( 90,340)( 91,339)( 92,338)( 93,333)( 94,336)( 95,335)( 96,334)
( 97,329)( 98,332)( 99,331)(100,330)(101,305)(102,308)(103,307)(104,306)
(105,301)(106,304)(107,303)(108,302)(109,317)(110,320)(111,319)(112,318)
(113,313)(114,316)(115,315)(116,314)(117,309)(118,312)(119,311)(120,310)
(121,465)(122,468)(123,467)(124,466)(125,461)(126,464)(127,463)(128,462)
(129,477)(130,480)(131,479)(132,478)(133,473)(134,476)(135,475)(136,474)
(137,469)(138,472)(139,471)(140,470)(141,445)(142,448)(143,447)(144,446)
(145,441)(146,444)(147,443)(148,442)(149,457)(150,460)(151,459)(152,458)
(153,453)(154,456)(155,455)(156,454)(157,449)(158,452)(159,451)(160,450)
(161,425)(162,428)(163,427)(164,426)(165,421)(166,424)(167,423)(168,422)
(169,437)(170,440)(171,439)(172,438)(173,433)(174,436)(175,435)(176,434)
(177,429)(178,432)(179,431)(180,430)(181,405)(182,408)(183,407)(184,406)
(185,401)(186,404)(187,403)(188,402)(189,417)(190,420)(191,419)(192,418)
(193,413)(194,416)(195,415)(196,414)(197,409)(198,412)(199,411)(200,410)
(201,385)(202,388)(203,387)(204,386)(205,381)(206,384)(207,383)(208,382)
(209,397)(210,400)(211,399)(212,398)(213,393)(214,396)(215,395)(216,394)
(217,389)(218,392)(219,391)(220,390)(221,365)(222,368)(223,367)(224,366)
(225,361)(226,364)(227,363)(228,362)(229,377)(230,380)(231,379)(232,378)
(233,373)(234,376)(235,375)(236,374)(237,369)(238,372)(239,371)(240,370);;
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*s0*s2*s1*s0*s2*s1*s0*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, 63)(  2, 64)(  3, 61)(  4, 62)(  5, 67)(  6, 68)(  7, 65)
(  8, 66)(  9, 71)( 10, 72)( 11, 69)( 12, 70)( 13, 75)( 14, 76)( 15, 73)
( 16, 74)( 17, 79)( 18, 80)( 19, 77)( 20, 78)( 21, 83)( 22, 84)( 23, 81)
( 24, 82)( 25, 87)( 26, 88)( 27, 85)( 28, 86)( 29, 91)( 30, 92)( 31, 89)
( 32, 90)( 33, 95)( 34, 96)( 35, 93)( 36, 94)( 37, 99)( 38,100)( 39, 97)
( 40, 98)( 41,103)( 42,104)( 43,101)( 44,102)( 45,107)( 46,108)( 47,105)
( 48,106)( 49,111)( 50,112)( 51,109)( 52,110)( 53,115)( 54,116)( 55,113)
( 56,114)( 57,119)( 58,120)( 59,117)( 60,118)(121,183)(122,184)(123,181)
(124,182)(125,187)(126,188)(127,185)(128,186)(129,191)(130,192)(131,189)
(132,190)(133,195)(134,196)(135,193)(136,194)(137,199)(138,200)(139,197)
(140,198)(141,203)(142,204)(143,201)(144,202)(145,207)(146,208)(147,205)
(148,206)(149,211)(150,212)(151,209)(152,210)(153,215)(154,216)(155,213)
(156,214)(157,219)(158,220)(159,217)(160,218)(161,223)(162,224)(163,221)
(164,222)(165,227)(166,228)(167,225)(168,226)(169,231)(170,232)(171,229)
(172,230)(173,235)(174,236)(175,233)(176,234)(177,239)(178,240)(179,237)
(180,238)(241,303)(242,304)(243,301)(244,302)(245,307)(246,308)(247,305)
(248,306)(249,311)(250,312)(251,309)(252,310)(253,315)(254,316)(255,313)
(256,314)(257,319)(258,320)(259,317)(260,318)(261,323)(262,324)(263,321)
(264,322)(265,327)(266,328)(267,325)(268,326)(269,331)(270,332)(271,329)
(272,330)(273,335)(274,336)(275,333)(276,334)(277,339)(278,340)(279,337)
(280,338)(281,343)(282,344)(283,341)(284,342)(285,347)(286,348)(287,345)
(288,346)(289,351)(290,352)(291,349)(292,350)(293,355)(294,356)(295,353)
(296,354)(297,359)(298,360)(299,357)(300,358)(361,423)(362,424)(363,421)
(364,422)(365,427)(366,428)(367,425)(368,426)(369,431)(370,432)(371,429)
(372,430)(373,435)(374,436)(375,433)(376,434)(377,439)(378,440)(379,437)
(380,438)(381,443)(382,444)(383,441)(384,442)(385,447)(386,448)(387,445)
(388,446)(389,451)(390,452)(391,449)(392,450)(393,455)(394,456)(395,453)
(396,454)(397,459)(398,460)(399,457)(400,458)(401,463)(402,464)(403,461)
(404,462)(405,467)(406,468)(407,465)(408,466)(409,471)(410,472)(411,469)
(412,470)(413,475)(414,476)(415,473)(416,474)(417,479)(418,480)(419,477)
(420,478);
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)(241,361)
(242,363)(243,362)(244,364)(245,377)(246,379)(247,378)(248,380)(249,373)
(250,375)(251,374)(252,376)(253,369)(254,371)(255,370)(256,372)(257,365)
(258,367)(259,366)(260,368)(261,401)(262,403)(263,402)(264,404)(265,417)
(266,419)(267,418)(268,420)(269,413)(270,415)(271,414)(272,416)(273,409)
(274,411)(275,410)(276,412)(277,405)(278,407)(279,406)(280,408)(281,381)
(282,383)(283,382)(284,384)(285,397)(286,399)(287,398)(288,400)(289,393)
(290,395)(291,394)(292,396)(293,389)(294,391)(295,390)(296,392)(297,385)
(298,387)(299,386)(300,388)(301,421)(302,423)(303,422)(304,424)(305,437)
(306,439)(307,438)(308,440)(309,433)(310,435)(311,434)(312,436)(313,429)
(314,431)(315,430)(316,432)(317,425)(318,427)(319,426)(320,428)(321,461)
(322,463)(323,462)(324,464)(325,477)(326,479)(327,478)(328,480)(329,473)
(330,475)(331,474)(332,476)(333,469)(334,471)(335,470)(336,472)(337,465)
(338,467)(339,466)(340,468)(341,441)(342,443)(343,442)(344,444)(345,457)
(346,459)(347,458)(348,460)(349,453)(350,455)(351,454)(352,456)(353,449)
(354,451)(355,450)(356,452)(357,445)(358,447)(359,446)(360,448);
s2 := Sym(480)!(  1,285)(  2,288)(  3,287)(  4,286)(  5,281)(  6,284)(  7,283)
(  8,282)(  9,297)( 10,300)( 11,299)( 12,298)( 13,293)( 14,296)( 15,295)
( 16,294)( 17,289)( 18,292)( 19,291)( 20,290)( 21,265)( 22,268)( 23,267)
( 24,266)( 25,261)( 26,264)( 27,263)( 28,262)( 29,277)( 30,280)( 31,279)
( 32,278)( 33,273)( 34,276)( 35,275)( 36,274)( 37,269)( 38,272)( 39,271)
( 40,270)( 41,245)( 42,248)( 43,247)( 44,246)( 45,241)( 46,244)( 47,243)
( 48,242)( 49,257)( 50,260)( 51,259)( 52,258)( 53,253)( 54,256)( 55,255)
( 56,254)( 57,249)( 58,252)( 59,251)( 60,250)( 61,345)( 62,348)( 63,347)
( 64,346)( 65,341)( 66,344)( 67,343)( 68,342)( 69,357)( 70,360)( 71,359)
( 72,358)( 73,353)( 74,356)( 75,355)( 76,354)( 77,349)( 78,352)( 79,351)
( 80,350)( 81,325)( 82,328)( 83,327)( 84,326)( 85,321)( 86,324)( 87,323)
( 88,322)( 89,337)( 90,340)( 91,339)( 92,338)( 93,333)( 94,336)( 95,335)
( 96,334)( 97,329)( 98,332)( 99,331)(100,330)(101,305)(102,308)(103,307)
(104,306)(105,301)(106,304)(107,303)(108,302)(109,317)(110,320)(111,319)
(112,318)(113,313)(114,316)(115,315)(116,314)(117,309)(118,312)(119,311)
(120,310)(121,465)(122,468)(123,467)(124,466)(125,461)(126,464)(127,463)
(128,462)(129,477)(130,480)(131,479)(132,478)(133,473)(134,476)(135,475)
(136,474)(137,469)(138,472)(139,471)(140,470)(141,445)(142,448)(143,447)
(144,446)(145,441)(146,444)(147,443)(148,442)(149,457)(150,460)(151,459)
(152,458)(153,453)(154,456)(155,455)(156,454)(157,449)(158,452)(159,451)
(160,450)(161,425)(162,428)(163,427)(164,426)(165,421)(166,424)(167,423)
(168,422)(169,437)(170,440)(171,439)(172,438)(173,433)(174,436)(175,435)
(176,434)(177,429)(178,432)(179,431)(180,430)(181,405)(182,408)(183,407)
(184,406)(185,401)(186,404)(187,403)(188,402)(189,417)(190,420)(191,419)
(192,418)(193,413)(194,416)(195,415)(196,414)(197,409)(198,412)(199,411)
(200,410)(201,385)(202,388)(203,387)(204,386)(205,381)(206,384)(207,383)
(208,382)(209,397)(210,400)(211,399)(212,398)(213,393)(214,396)(215,395)
(216,394)(217,389)(218,392)(219,391)(220,390)(221,365)(222,368)(223,367)
(224,366)(225,361)(226,364)(227,363)(228,362)(229,377)(230,380)(231,379)
(232,378)(233,373)(234,376)(235,375)(236,374)(237,369)(238,372)(239,371)
(240,370);
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*s0*s2*s1*s0*s2*s1*s0*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