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# Polytope of Type {6,40}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,40}*1920d
if this polytope has a name.
Group : SmallGroup(1920,238620)
Rank : 3
Schlafli Type : {6,40}
Number of vertices, edges, etc : 24, 480, 160
Order of s0s1s2 : 120
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 : {6,20}*960e
4-fold quotients : {6,40}*480, {6,20}*480c
5-fold quotients : {6,8}*384g
8-fold quotients : {6,20}*240a, {6,20}*240b
10-fold quotients : {6,4}*192b
12-fold quotients : {2,40}*160
16-fold quotients : {6,10}*120
20-fold quotients : {6,8}*96, {6,4}*96
24-fold quotients : {2,20}*80
40-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {2,10}*40
60-fold quotients : {2,8}*32
80-fold quotients : {3,4}*24, {6,2}*24
96-fold quotients : {2,5}*20
120-fold quotients : {2,4}*16
160-fold quotients : {3,2}*12
240-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 21, 41)( 22, 42)( 23, 44)
( 24, 43)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 49)( 30, 50)( 31, 52)
( 32, 51)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 37, 57)( 38, 58)( 39, 60)
( 40, 59)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 81,101)( 82,102)
( 83,104)( 84,103)( 85,105)( 86,106)( 87,108)( 88,107)( 89,109)( 90,110)
( 91,112)( 92,111)( 93,113)( 94,114)( 95,116)( 96,115)( 97,117)( 98,118)
( 99,120)(100,119)(123,124)(127,128)(131,132)(135,136)(139,140)(141,161)
(142,162)(143,164)(144,163)(145,165)(146,166)(147,168)(148,167)(149,169)
(150,170)(151,172)(152,171)(153,173)(154,174)(155,176)(156,175)(157,177)
(158,178)(159,180)(160,179)(183,184)(187,188)(191,192)(195,196)(199,200)
(201,221)(202,222)(203,224)(204,223)(205,225)(206,226)(207,228)(208,227)
(209,229)(210,230)(211,232)(212,231)(213,233)(214,234)(215,236)(216,235)
(217,237)(218,238)(219,240)(220,239)(243,244)(247,248)(251,252)(255,256)
(259,260)(261,281)(262,282)(263,284)(264,283)(265,285)(266,286)(267,288)
(268,287)(269,289)(270,290)(271,292)(272,291)(273,293)(274,294)(275,296)
(276,295)(277,297)(278,298)(279,300)(280,299)(303,304)(307,308)(311,312)
(315,316)(319,320)(321,341)(322,342)(323,344)(324,343)(325,345)(326,346)
(327,348)(328,347)(329,349)(330,350)(331,352)(332,351)(333,353)(334,354)
(335,356)(336,355)(337,357)(338,358)(339,360)(340,359)(363,364)(367,368)
(371,372)(375,376)(379,380)(381,401)(382,402)(383,404)(384,403)(385,405)
(386,406)(387,408)(388,407)(389,409)(390,410)(391,412)(392,411)(393,413)
(394,414)(395,416)(396,415)(397,417)(398,418)(399,420)(400,419)(423,424)
(427,428)(431,432)(435,436)(439,440)(441,461)(442,462)(443,464)(444,463)
(445,465)(446,466)(447,468)(448,467)(449,469)(450,470)(451,472)(452,471)
(453,473)(454,474)(455,476)(456,475)(457,477)(458,478)(459,480)(460,479);;
s1 := (  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 37)(  6, 40)(  7, 39)(  8, 38)
(  9, 33)( 10, 36)( 11, 35)( 12, 34)( 13, 29)( 14, 32)( 15, 31)( 16, 30)
( 17, 25)( 18, 28)( 19, 27)( 20, 26)( 42, 44)( 45, 57)( 46, 60)( 47, 59)
( 48, 58)( 49, 53)( 50, 56)( 51, 55)( 52, 54)( 61, 81)( 62, 84)( 63, 83)
( 64, 82)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69, 93)( 70, 96)( 71, 95)
( 72, 94)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)( 79, 87)
( 80, 86)(102,104)(105,117)(106,120)(107,119)(108,118)(109,113)(110,116)
(111,115)(112,114)(121,201)(122,204)(123,203)(124,202)(125,217)(126,220)
(127,219)(128,218)(129,213)(130,216)(131,215)(132,214)(133,209)(134,212)
(135,211)(136,210)(137,205)(138,208)(139,207)(140,206)(141,181)(142,184)
(143,183)(144,182)(145,197)(146,200)(147,199)(148,198)(149,193)(150,196)
(151,195)(152,194)(153,189)(154,192)(155,191)(156,190)(157,185)(158,188)
(159,187)(160,186)(161,221)(162,224)(163,223)(164,222)(165,237)(166,240)
(167,239)(168,238)(169,233)(170,236)(171,235)(172,234)(173,229)(174,232)
(175,231)(176,230)(177,225)(178,228)(179,227)(180,226)(241,381)(242,384)
(243,383)(244,382)(245,397)(246,400)(247,399)(248,398)(249,393)(250,396)
(251,395)(252,394)(253,389)(254,392)(255,391)(256,390)(257,385)(258,388)
(259,387)(260,386)(261,361)(262,364)(263,363)(264,362)(265,377)(266,380)
(267,379)(268,378)(269,373)(270,376)(271,375)(272,374)(273,369)(274,372)
(275,371)(276,370)(277,365)(278,368)(279,367)(280,366)(281,401)(282,404)
(283,403)(284,402)(285,417)(286,420)(287,419)(288,418)(289,413)(290,416)
(291,415)(292,414)(293,409)(294,412)(295,411)(296,410)(297,405)(298,408)
(299,407)(300,406)(301,441)(302,444)(303,443)(304,442)(305,457)(306,460)
(307,459)(308,458)(309,453)(310,456)(311,455)(312,454)(313,449)(314,452)
(315,451)(316,450)(317,445)(318,448)(319,447)(320,446)(321,421)(322,424)
(323,423)(324,422)(325,437)(326,440)(327,439)(328,438)(329,433)(330,436)
(331,435)(332,434)(333,429)(334,432)(335,431)(336,430)(337,425)(338,428)
(339,427)(340,426)(341,461)(342,464)(343,463)(344,462)(345,477)(346,480)
(347,479)(348,478)(349,473)(350,476)(351,475)(352,474)(353,469)(354,472)
(355,471)(356,470)(357,465)(358,468)(359,467)(360,466);;
s2 := (  1,246)(  2,245)(  3,248)(  4,247)(  5,242)(  6,241)(  7,244)(  8,243)
(  9,258)( 10,257)( 11,260)( 12,259)( 13,254)( 14,253)( 15,256)( 16,255)
( 17,250)( 18,249)( 19,252)( 20,251)( 21,266)( 22,265)( 23,268)( 24,267)
( 25,262)( 26,261)( 27,264)( 28,263)( 29,278)( 30,277)( 31,280)( 32,279)
( 33,274)( 34,273)( 35,276)( 36,275)( 37,270)( 38,269)( 39,272)( 40,271)
( 41,286)( 42,285)( 43,288)( 44,287)( 45,282)( 46,281)( 47,284)( 48,283)
( 49,298)( 50,297)( 51,300)( 52,299)( 53,294)( 54,293)( 55,296)( 56,295)
( 57,290)( 58,289)( 59,292)( 60,291)( 61,306)( 62,305)( 63,308)( 64,307)
( 65,302)( 66,301)( 67,304)( 68,303)( 69,318)( 70,317)( 71,320)( 72,319)
( 73,314)( 74,313)( 75,316)( 76,315)( 77,310)( 78,309)( 79,312)( 80,311)
( 81,326)( 82,325)( 83,328)( 84,327)( 85,322)( 86,321)( 87,324)( 88,323)
( 89,338)( 90,337)( 91,340)( 92,339)( 93,334)( 94,333)( 95,336)( 96,335)
( 97,330)( 98,329)( 99,332)(100,331)(101,346)(102,345)(103,348)(104,347)
(105,342)(106,341)(107,344)(108,343)(109,358)(110,357)(111,360)(112,359)
(113,354)(114,353)(115,356)(116,355)(117,350)(118,349)(119,352)(120,351)
(121,426)(122,425)(123,428)(124,427)(125,422)(126,421)(127,424)(128,423)
(129,438)(130,437)(131,440)(132,439)(133,434)(134,433)(135,436)(136,435)
(137,430)(138,429)(139,432)(140,431)(141,446)(142,445)(143,448)(144,447)
(145,442)(146,441)(147,444)(148,443)(149,458)(150,457)(151,460)(152,459)
(153,454)(154,453)(155,456)(156,455)(157,450)(158,449)(159,452)(160,451)
(161,466)(162,465)(163,468)(164,467)(165,462)(166,461)(167,464)(168,463)
(169,478)(170,477)(171,480)(172,479)(173,474)(174,473)(175,476)(176,475)
(177,470)(178,469)(179,472)(180,471)(181,366)(182,365)(183,368)(184,367)
(185,362)(186,361)(187,364)(188,363)(189,378)(190,377)(191,380)(192,379)
(193,374)(194,373)(195,376)(196,375)(197,370)(198,369)(199,372)(200,371)
(201,386)(202,385)(203,388)(204,387)(205,382)(206,381)(207,384)(208,383)
(209,398)(210,397)(211,400)(212,399)(213,394)(214,393)(215,396)(216,395)
(217,390)(218,389)(219,392)(220,391)(221,406)(222,405)(223,408)(224,407)
(225,402)(226,401)(227,404)(228,403)(229,418)(230,417)(231,420)(232,419)
(233,414)(234,413)(235,416)(236,415)(237,410)(238,409)(239,412)(240,411);;
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*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*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 ];;
poly := F / rels;;

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

References : None.
to this polytope