Polytope of Type {6,165}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,165}*1980
if this polytope has a name.
Group : SmallGroup(1980,103)
Rank : 3
Schlafli Type : {6,165}
Number of vertices, edges, etc : 6, 495, 165
Order of s0s1s2 : 330
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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) :
3-fold quotients : {2,165}*660
5-fold quotients : {6,33}*396
9-fold quotients : {2,55}*220
11-fold quotients : {6,15}*180
15-fold quotients : {2,33}*132
33-fold quotients : {2,15}*60
45-fold quotients : {2,11}*44
55-fold quotients : {6,3}*36
99-fold quotients : {2,5}*20
165-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)( 63,118)
( 64,119)( 65,120)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)( 71,126)
( 72,127)( 73,128)( 74,129)( 75,130)( 76,131)( 77,132)( 78,133)( 79,134)
( 80,135)( 81,136)( 82,137)( 83,138)( 84,139)( 85,140)( 86,141)( 87,142)
( 88,143)( 89,144)( 90,145)( 91,146)( 92,147)( 93,148)( 94,149)( 95,150)
( 96,151)( 97,152)( 98,153)( 99,154)(100,155)(101,156)(102,157)(103,158)
(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(221,276)
(222,277)(223,278)(224,279)(225,280)(226,281)(227,282)(228,283)(229,284)
(230,285)(231,286)(232,287)(233,288)(234,289)(235,290)(236,291)(237,292)
(238,293)(239,294)(240,295)(241,296)(242,297)(243,298)(244,299)(245,300)
(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)(253,308)
(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315)(261,316)
(262,317)(263,318)(264,319)(265,320)(266,321)(267,322)(268,323)(269,324)
(270,325)(271,326)(272,327)(273,328)(274,329)(275,330)(386,441)(387,442)
(388,443)(389,444)(390,445)(391,446)(392,447)(393,448)(394,449)(395,450)
(396,451)(397,452)(398,453)(399,454)(400,455)(401,456)(402,457)(403,458)
(404,459)(405,460)(406,461)(407,462)(408,463)(409,464)(410,465)(411,466)
(412,467)(413,468)(414,469)(415,470)(416,471)(417,472)(418,473)(419,474)
(420,475)(421,476)(422,477)(423,478)(424,479)(425,480)(426,481)(427,482)
(428,483)(429,484)(430,485)(431,486)(432,487)(433,488)(434,489)(435,490)
(436,491)(437,492)(438,493)(439,494)(440,495);;
s1 := ( 1, 56)( 2, 66)( 3, 65)( 4, 64)( 5, 63)( 6, 62)( 7, 61)( 8, 60)
( 9, 59)( 10, 58)( 11, 57)( 12,100)( 13,110)( 14,109)( 15,108)( 16,107)
( 17,106)( 18,105)( 19,104)( 20,103)( 21,102)( 22,101)( 23, 89)( 24, 99)
( 25, 98)( 26, 97)( 27, 96)( 28, 95)( 29, 94)( 30, 93)( 31, 92)( 32, 91)
( 33, 90)( 34, 78)( 35, 88)( 36, 87)( 37, 86)( 38, 85)( 39, 84)( 40, 83)
( 41, 82)( 42, 81)( 43, 80)( 44, 79)( 45, 67)( 46, 77)( 47, 76)( 48, 75)
( 49, 74)( 50, 73)( 51, 72)( 52, 71)( 53, 70)( 54, 69)( 55, 68)(112,121)
(113,120)(114,119)(115,118)(116,117)(122,155)(123,165)(124,164)(125,163)
(126,162)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)(133,144)
(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)
(142,146)(143,145)(166,386)(167,396)(168,395)(169,394)(170,393)(171,392)
(172,391)(173,390)(174,389)(175,388)(176,387)(177,430)(178,440)(179,439)
(180,438)(181,437)(182,436)(183,435)(184,434)(185,433)(186,432)(187,431)
(188,419)(189,429)(190,428)(191,427)(192,426)(193,425)(194,424)(195,423)
(196,422)(197,421)(198,420)(199,408)(200,418)(201,417)(202,416)(203,415)
(204,414)(205,413)(206,412)(207,411)(208,410)(209,409)(210,397)(211,407)
(212,406)(213,405)(214,404)(215,403)(216,402)(217,401)(218,400)(219,399)
(220,398)(221,331)(222,341)(223,340)(224,339)(225,338)(226,337)(227,336)
(228,335)(229,334)(230,333)(231,332)(232,375)(233,385)(234,384)(235,383)
(236,382)(237,381)(238,380)(239,379)(240,378)(241,377)(242,376)(243,364)
(244,374)(245,373)(246,372)(247,371)(248,370)(249,369)(250,368)(251,367)
(252,366)(253,365)(254,353)(255,363)(256,362)(257,361)(258,360)(259,359)
(260,358)(261,357)(262,356)(263,355)(264,354)(265,342)(266,352)(267,351)
(268,350)(269,349)(270,348)(271,347)(272,346)(273,345)(274,344)(275,343)
(276,441)(277,451)(278,450)(279,449)(280,448)(281,447)(282,446)(283,445)
(284,444)(285,443)(286,442)(287,485)(288,495)(289,494)(290,493)(291,492)
(292,491)(293,490)(294,489)(295,488)(296,487)(297,486)(298,474)(299,484)
(300,483)(301,482)(302,481)(303,480)(304,479)(305,478)(306,477)(307,476)
(308,475)(309,463)(310,473)(311,472)(312,471)(313,470)(314,469)(315,468)
(316,467)(317,466)(318,465)(319,464)(320,452)(321,462)(322,461)(323,460)
(324,459)(325,458)(326,457)(327,456)(328,455)(329,454)(330,453);;
s2 := ( 1,178)( 2,177)( 3,187)( 4,186)( 5,185)( 6,184)( 7,183)( 8,182)
( 9,181)( 10,180)( 11,179)( 12,167)( 13,166)( 14,176)( 15,175)( 16,174)
( 17,173)( 18,172)( 19,171)( 20,170)( 21,169)( 22,168)( 23,211)( 24,210)
( 25,220)( 26,219)( 27,218)( 28,217)( 29,216)( 30,215)( 31,214)( 32,213)
( 33,212)( 34,200)( 35,199)( 36,209)( 37,208)( 38,207)( 39,206)( 40,205)
( 41,204)( 42,203)( 43,202)( 44,201)( 45,189)( 46,188)( 47,198)( 48,197)
( 49,196)( 50,195)( 51,194)( 52,193)( 53,192)( 54,191)( 55,190)( 56,288)
( 57,287)( 58,297)( 59,296)( 60,295)( 61,294)( 62,293)( 63,292)( 64,291)
( 65,290)( 66,289)( 67,277)( 68,276)( 69,286)( 70,285)( 71,284)( 72,283)
( 73,282)( 74,281)( 75,280)( 76,279)( 77,278)( 78,321)( 79,320)( 80,330)
( 81,329)( 82,328)( 83,327)( 84,326)( 85,325)( 86,324)( 87,323)( 88,322)
( 89,310)( 90,309)( 91,319)( 92,318)( 93,317)( 94,316)( 95,315)( 96,314)
( 97,313)( 98,312)( 99,311)(100,299)(101,298)(102,308)(103,307)(104,306)
(105,305)(106,304)(107,303)(108,302)(109,301)(110,300)(111,233)(112,232)
(113,242)(114,241)(115,240)(116,239)(117,238)(118,237)(119,236)(120,235)
(121,234)(122,222)(123,221)(124,231)(125,230)(126,229)(127,228)(128,227)
(129,226)(130,225)(131,224)(132,223)(133,266)(134,265)(135,275)(136,274)
(137,273)(138,272)(139,271)(140,270)(141,269)(142,268)(143,267)(144,255)
(145,254)(146,264)(147,263)(148,262)(149,261)(150,260)(151,259)(152,258)
(153,257)(154,256)(155,244)(156,243)(157,253)(158,252)(159,251)(160,250)
(161,249)(162,248)(163,247)(164,246)(165,245)(331,343)(332,342)(333,352)
(334,351)(335,350)(336,349)(337,348)(338,347)(339,346)(340,345)(341,344)
(353,376)(354,375)(355,385)(356,384)(357,383)(358,382)(359,381)(360,380)
(361,379)(362,378)(363,377)(364,365)(366,374)(367,373)(368,372)(369,371)
(386,453)(387,452)(388,462)(389,461)(390,460)(391,459)(392,458)(393,457)
(394,456)(395,455)(396,454)(397,442)(398,441)(399,451)(400,450)(401,449)
(402,448)(403,447)(404,446)(405,445)(406,444)(407,443)(408,486)(409,485)
(410,495)(411,494)(412,493)(413,492)(414,491)(415,490)(416,489)(417,488)
(418,487)(419,475)(420,474)(421,484)(422,483)(423,482)(424,481)(425,480)
(426,479)(427,478)(428,477)(429,476)(430,464)(431,463)(432,473)(433,472)
(434,471)(435,470)(436,469)(437,468)(438,467)(439,466)(440,465);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(495)!( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)
( 63,118)( 64,119)( 65,120)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)
( 71,126)( 72,127)( 73,128)( 74,129)( 75,130)( 76,131)( 77,132)( 78,133)
( 79,134)( 80,135)( 81,136)( 82,137)( 83,138)( 84,139)( 85,140)( 86,141)
( 87,142)( 88,143)( 89,144)( 90,145)( 91,146)( 92,147)( 93,148)( 94,149)
( 95,150)( 96,151)( 97,152)( 98,153)( 99,154)(100,155)(101,156)(102,157)
(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)
(221,276)(222,277)(223,278)(224,279)(225,280)(226,281)(227,282)(228,283)
(229,284)(230,285)(231,286)(232,287)(233,288)(234,289)(235,290)(236,291)
(237,292)(238,293)(239,294)(240,295)(241,296)(242,297)(243,298)(244,299)
(245,300)(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)
(253,308)(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315)
(261,316)(262,317)(263,318)(264,319)(265,320)(266,321)(267,322)(268,323)
(269,324)(270,325)(271,326)(272,327)(273,328)(274,329)(275,330)(386,441)
(387,442)(388,443)(389,444)(390,445)(391,446)(392,447)(393,448)(394,449)
(395,450)(396,451)(397,452)(398,453)(399,454)(400,455)(401,456)(402,457)
(403,458)(404,459)(405,460)(406,461)(407,462)(408,463)(409,464)(410,465)
(411,466)(412,467)(413,468)(414,469)(415,470)(416,471)(417,472)(418,473)
(419,474)(420,475)(421,476)(422,477)(423,478)(424,479)(425,480)(426,481)
(427,482)(428,483)(429,484)(430,485)(431,486)(432,487)(433,488)(434,489)
(435,490)(436,491)(437,492)(438,493)(439,494)(440,495);
s1 := Sym(495)!( 1, 56)( 2, 66)( 3, 65)( 4, 64)( 5, 63)( 6, 62)( 7, 61)
( 8, 60)( 9, 59)( 10, 58)( 11, 57)( 12,100)( 13,110)( 14,109)( 15,108)
( 16,107)( 17,106)( 18,105)( 19,104)( 20,103)( 21,102)( 22,101)( 23, 89)
( 24, 99)( 25, 98)( 26, 97)( 27, 96)( 28, 95)( 29, 94)( 30, 93)( 31, 92)
( 32, 91)( 33, 90)( 34, 78)( 35, 88)( 36, 87)( 37, 86)( 38, 85)( 39, 84)
( 40, 83)( 41, 82)( 42, 81)( 43, 80)( 44, 79)( 45, 67)( 46, 77)( 47, 76)
( 48, 75)( 49, 74)( 50, 73)( 51, 72)( 52, 71)( 53, 70)( 54, 69)( 55, 68)
(112,121)(113,120)(114,119)(115,118)(116,117)(122,155)(123,165)(124,164)
(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)
(133,144)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)
(141,147)(142,146)(143,145)(166,386)(167,396)(168,395)(169,394)(170,393)
(171,392)(172,391)(173,390)(174,389)(175,388)(176,387)(177,430)(178,440)
(179,439)(180,438)(181,437)(182,436)(183,435)(184,434)(185,433)(186,432)
(187,431)(188,419)(189,429)(190,428)(191,427)(192,426)(193,425)(194,424)
(195,423)(196,422)(197,421)(198,420)(199,408)(200,418)(201,417)(202,416)
(203,415)(204,414)(205,413)(206,412)(207,411)(208,410)(209,409)(210,397)
(211,407)(212,406)(213,405)(214,404)(215,403)(216,402)(217,401)(218,400)
(219,399)(220,398)(221,331)(222,341)(223,340)(224,339)(225,338)(226,337)
(227,336)(228,335)(229,334)(230,333)(231,332)(232,375)(233,385)(234,384)
(235,383)(236,382)(237,381)(238,380)(239,379)(240,378)(241,377)(242,376)
(243,364)(244,374)(245,373)(246,372)(247,371)(248,370)(249,369)(250,368)
(251,367)(252,366)(253,365)(254,353)(255,363)(256,362)(257,361)(258,360)
(259,359)(260,358)(261,357)(262,356)(263,355)(264,354)(265,342)(266,352)
(267,351)(268,350)(269,349)(270,348)(271,347)(272,346)(273,345)(274,344)
(275,343)(276,441)(277,451)(278,450)(279,449)(280,448)(281,447)(282,446)
(283,445)(284,444)(285,443)(286,442)(287,485)(288,495)(289,494)(290,493)
(291,492)(292,491)(293,490)(294,489)(295,488)(296,487)(297,486)(298,474)
(299,484)(300,483)(301,482)(302,481)(303,480)(304,479)(305,478)(306,477)
(307,476)(308,475)(309,463)(310,473)(311,472)(312,471)(313,470)(314,469)
(315,468)(316,467)(317,466)(318,465)(319,464)(320,452)(321,462)(322,461)
(323,460)(324,459)(325,458)(326,457)(327,456)(328,455)(329,454)(330,453);
s2 := Sym(495)!( 1,178)( 2,177)( 3,187)( 4,186)( 5,185)( 6,184)( 7,183)
( 8,182)( 9,181)( 10,180)( 11,179)( 12,167)( 13,166)( 14,176)( 15,175)
( 16,174)( 17,173)( 18,172)( 19,171)( 20,170)( 21,169)( 22,168)( 23,211)
( 24,210)( 25,220)( 26,219)( 27,218)( 28,217)( 29,216)( 30,215)( 31,214)
( 32,213)( 33,212)( 34,200)( 35,199)( 36,209)( 37,208)( 38,207)( 39,206)
( 40,205)( 41,204)( 42,203)( 43,202)( 44,201)( 45,189)( 46,188)( 47,198)
( 48,197)( 49,196)( 50,195)( 51,194)( 52,193)( 53,192)( 54,191)( 55,190)
( 56,288)( 57,287)( 58,297)( 59,296)( 60,295)( 61,294)( 62,293)( 63,292)
( 64,291)( 65,290)( 66,289)( 67,277)( 68,276)( 69,286)( 70,285)( 71,284)
( 72,283)( 73,282)( 74,281)( 75,280)( 76,279)( 77,278)( 78,321)( 79,320)
( 80,330)( 81,329)( 82,328)( 83,327)( 84,326)( 85,325)( 86,324)( 87,323)
( 88,322)( 89,310)( 90,309)( 91,319)( 92,318)( 93,317)( 94,316)( 95,315)
( 96,314)( 97,313)( 98,312)( 99,311)(100,299)(101,298)(102,308)(103,307)
(104,306)(105,305)(106,304)(107,303)(108,302)(109,301)(110,300)(111,233)
(112,232)(113,242)(114,241)(115,240)(116,239)(117,238)(118,237)(119,236)
(120,235)(121,234)(122,222)(123,221)(124,231)(125,230)(126,229)(127,228)
(128,227)(129,226)(130,225)(131,224)(132,223)(133,266)(134,265)(135,275)
(136,274)(137,273)(138,272)(139,271)(140,270)(141,269)(142,268)(143,267)
(144,255)(145,254)(146,264)(147,263)(148,262)(149,261)(150,260)(151,259)
(152,258)(153,257)(154,256)(155,244)(156,243)(157,253)(158,252)(159,251)
(160,250)(161,249)(162,248)(163,247)(164,246)(165,245)(331,343)(332,342)
(333,352)(334,351)(335,350)(336,349)(337,348)(338,347)(339,346)(340,345)
(341,344)(353,376)(354,375)(355,385)(356,384)(357,383)(358,382)(359,381)
(360,380)(361,379)(362,378)(363,377)(364,365)(366,374)(367,373)(368,372)
(369,371)(386,453)(387,452)(388,462)(389,461)(390,460)(391,459)(392,458)
(393,457)(394,456)(395,455)(396,454)(397,442)(398,441)(399,451)(400,450)
(401,449)(402,448)(403,447)(404,446)(405,445)(406,444)(407,443)(408,486)
(409,485)(410,495)(411,494)(412,493)(413,492)(414,491)(415,490)(416,489)
(417,488)(418,487)(419,475)(420,474)(421,484)(422,483)(423,482)(424,481)
(425,480)(426,479)(427,478)(428,477)(429,476)(430,464)(431,463)(432,473)
(433,472)(434,471)(435,470)(436,469)(437,468)(438,467)(439,466)(440,465);
poly := sub<Sym(495)|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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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