Polytope of Type {6,166}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,166}*1992
Also Known As : {6,166|2}. if this polytope has another name.
Group : SmallGroup(1992,34)
Rank : 3
Schlafli Type : {6,166}
Number of vertices, edges, etc : 6, 498, 166
Order of s0s1s2 : 498
Order of s0s1s2s1 : 2
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 : {2,166}*664
6-fold quotients : {2,83}*332
83-fold quotients : {6,2}*24
166-fold quotients : {3,2}*12
249-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 84,167)( 85,168)( 86,169)( 87,170)( 88,171)( 89,172)( 90,173)( 91,174)
( 92,175)( 93,176)( 94,177)( 95,178)( 96,179)( 97,180)( 98,181)( 99,182)
(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)
(108,191)(109,192)(110,193)(111,194)(112,195)(113,196)(114,197)(115,198)
(116,199)(117,200)(118,201)(119,202)(120,203)(121,204)(122,205)(123,206)
(124,207)(125,208)(126,209)(127,210)(128,211)(129,212)(130,213)(131,214)
(132,215)(133,216)(134,217)(135,218)(136,219)(137,220)(138,221)(139,222)
(140,223)(141,224)(142,225)(143,226)(144,227)(145,228)(146,229)(147,230)
(148,231)(149,232)(150,233)(151,234)(152,235)(153,236)(154,237)(155,238)
(156,239)(157,240)(158,241)(159,242)(160,243)(161,244)(162,245)(163,246)
(164,247)(165,248)(166,249)(333,416)(334,417)(335,418)(336,419)(337,420)
(338,421)(339,422)(340,423)(341,424)(342,425)(343,426)(344,427)(345,428)
(346,429)(347,430)(348,431)(349,432)(350,433)(351,434)(352,435)(353,436)
(354,437)(355,438)(356,439)(357,440)(358,441)(359,442)(360,443)(361,444)
(362,445)(363,446)(364,447)(365,448)(366,449)(367,450)(368,451)(369,452)
(370,453)(371,454)(372,455)(373,456)(374,457)(375,458)(376,459)(377,460)
(378,461)(379,462)(380,463)(381,464)(382,465)(383,466)(384,467)(385,468)
(386,469)(387,470)(388,471)(389,472)(390,473)(391,474)(392,475)(393,476)
(394,477)(395,478)(396,479)(397,480)(398,481)(399,482)(400,483)(401,484)
(402,485)(403,486)(404,487)(405,488)(406,489)(407,490)(408,491)(409,492)
(410,493)(411,494)(412,495)(413,496)(414,497)(415,498);;
s1 := ( 1, 84)( 2,166)( 3,165)( 4,164)( 5,163)( 6,162)( 7,161)( 8,160)
( 9,159)( 10,158)( 11,157)( 12,156)( 13,155)( 14,154)( 15,153)( 16,152)
( 17,151)( 18,150)( 19,149)( 20,148)( 21,147)( 22,146)( 23,145)( 24,144)
( 25,143)( 26,142)( 27,141)( 28,140)( 29,139)( 30,138)( 31,137)( 32,136)
( 33,135)( 34,134)( 35,133)( 36,132)( 37,131)( 38,130)( 39,129)( 40,128)
( 41,127)( 42,126)( 43,125)( 44,124)( 45,123)( 46,122)( 47,121)( 48,120)
( 49,119)( 50,118)( 51,117)( 52,116)( 53,115)( 54,114)( 55,113)( 56,112)
( 57,111)( 58,110)( 59,109)( 60,108)( 61,107)( 62,106)( 63,105)( 64,104)
( 65,103)( 66,102)( 67,101)( 68,100)( 69, 99)( 70, 98)( 71, 97)( 72, 96)
( 73, 95)( 74, 94)( 75, 93)( 76, 92)( 77, 91)( 78, 90)( 79, 89)( 80, 88)
( 81, 87)( 82, 86)( 83, 85)(168,249)(169,248)(170,247)(171,246)(172,245)
(173,244)(174,243)(175,242)(176,241)(177,240)(178,239)(179,238)(180,237)
(181,236)(182,235)(183,234)(184,233)(185,232)(186,231)(187,230)(188,229)
(189,228)(190,227)(191,226)(192,225)(193,224)(194,223)(195,222)(196,221)
(197,220)(198,219)(199,218)(200,217)(201,216)(202,215)(203,214)(204,213)
(205,212)(206,211)(207,210)(208,209)(250,333)(251,415)(252,414)(253,413)
(254,412)(255,411)(256,410)(257,409)(258,408)(259,407)(260,406)(261,405)
(262,404)(263,403)(264,402)(265,401)(266,400)(267,399)(268,398)(269,397)
(270,396)(271,395)(272,394)(273,393)(274,392)(275,391)(276,390)(277,389)
(278,388)(279,387)(280,386)(281,385)(282,384)(283,383)(284,382)(285,381)
(286,380)(287,379)(288,378)(289,377)(290,376)(291,375)(292,374)(293,373)
(294,372)(295,371)(296,370)(297,369)(298,368)(299,367)(300,366)(301,365)
(302,364)(303,363)(304,362)(305,361)(306,360)(307,359)(308,358)(309,357)
(310,356)(311,355)(312,354)(313,353)(314,352)(315,351)(316,350)(317,349)
(318,348)(319,347)(320,346)(321,345)(322,344)(323,343)(324,342)(325,341)
(326,340)(327,339)(328,338)(329,337)(330,336)(331,335)(332,334)(417,498)
(418,497)(419,496)(420,495)(421,494)(422,493)(423,492)(424,491)(425,490)
(426,489)(427,488)(428,487)(429,486)(430,485)(431,484)(432,483)(433,482)
(434,481)(435,480)(436,479)(437,478)(438,477)(439,476)(440,475)(441,474)
(442,473)(443,472)(444,471)(445,470)(446,469)(447,468)(448,467)(449,466)
(450,465)(451,464)(452,463)(453,462)(454,461)(455,460)(456,459)(457,458);;
s2 := ( 1,251)( 2,250)( 3,332)( 4,331)( 5,330)( 6,329)( 7,328)( 8,327)
( 9,326)( 10,325)( 11,324)( 12,323)( 13,322)( 14,321)( 15,320)( 16,319)
( 17,318)( 18,317)( 19,316)( 20,315)( 21,314)( 22,313)( 23,312)( 24,311)
( 25,310)( 26,309)( 27,308)( 28,307)( 29,306)( 30,305)( 31,304)( 32,303)
( 33,302)( 34,301)( 35,300)( 36,299)( 37,298)( 38,297)( 39,296)( 40,295)
( 41,294)( 42,293)( 43,292)( 44,291)( 45,290)( 46,289)( 47,288)( 48,287)
( 49,286)( 50,285)( 51,284)( 52,283)( 53,282)( 54,281)( 55,280)( 56,279)
( 57,278)( 58,277)( 59,276)( 60,275)( 61,274)( 62,273)( 63,272)( 64,271)
( 65,270)( 66,269)( 67,268)( 68,267)( 69,266)( 70,265)( 71,264)( 72,263)
( 73,262)( 74,261)( 75,260)( 76,259)( 77,258)( 78,257)( 79,256)( 80,255)
( 81,254)( 82,253)( 83,252)( 84,334)( 85,333)( 86,415)( 87,414)( 88,413)
( 89,412)( 90,411)( 91,410)( 92,409)( 93,408)( 94,407)( 95,406)( 96,405)
( 97,404)( 98,403)( 99,402)(100,401)(101,400)(102,399)(103,398)(104,397)
(105,396)(106,395)(107,394)(108,393)(109,392)(110,391)(111,390)(112,389)
(113,388)(114,387)(115,386)(116,385)(117,384)(118,383)(119,382)(120,381)
(121,380)(122,379)(123,378)(124,377)(125,376)(126,375)(127,374)(128,373)
(129,372)(130,371)(131,370)(132,369)(133,368)(134,367)(135,366)(136,365)
(137,364)(138,363)(139,362)(140,361)(141,360)(142,359)(143,358)(144,357)
(145,356)(146,355)(147,354)(148,353)(149,352)(150,351)(151,350)(152,349)
(153,348)(154,347)(155,346)(156,345)(157,344)(158,343)(159,342)(160,341)
(161,340)(162,339)(163,338)(164,337)(165,336)(166,335)(167,417)(168,416)
(169,498)(170,497)(171,496)(172,495)(173,494)(174,493)(175,492)(176,491)
(177,490)(178,489)(179,488)(180,487)(181,486)(182,485)(183,484)(184,483)
(185,482)(186,481)(187,480)(188,479)(189,478)(190,477)(191,476)(192,475)
(193,474)(194,473)(195,472)(196,471)(197,470)(198,469)(199,468)(200,467)
(201,466)(202,465)(203,464)(204,463)(205,462)(206,461)(207,460)(208,459)
(209,458)(210,457)(211,456)(212,455)(213,454)(214,453)(215,452)(216,451)
(217,450)(218,449)(219,448)(220,447)(221,446)(222,445)(223,444)(224,443)
(225,442)(226,441)(227,440)(228,439)(229,438)(230,437)(231,436)(232,435)
(233,434)(234,433)(235,432)(236,431)(237,430)(238,429)(239,428)(240,427)
(241,426)(242,425)(243,424)(244,423)(245,422)(246,421)(247,420)(248,419)
(249,418);;
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*s2*s1*s0*s1*s2*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*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(498)!( 84,167)( 85,168)( 86,169)( 87,170)( 88,171)( 89,172)( 90,173)
( 91,174)( 92,175)( 93,176)( 94,177)( 95,178)( 96,179)( 97,180)( 98,181)
( 99,182)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)
(107,190)(108,191)(109,192)(110,193)(111,194)(112,195)(113,196)(114,197)
(115,198)(116,199)(117,200)(118,201)(119,202)(120,203)(121,204)(122,205)
(123,206)(124,207)(125,208)(126,209)(127,210)(128,211)(129,212)(130,213)
(131,214)(132,215)(133,216)(134,217)(135,218)(136,219)(137,220)(138,221)
(139,222)(140,223)(141,224)(142,225)(143,226)(144,227)(145,228)(146,229)
(147,230)(148,231)(149,232)(150,233)(151,234)(152,235)(153,236)(154,237)
(155,238)(156,239)(157,240)(158,241)(159,242)(160,243)(161,244)(162,245)
(163,246)(164,247)(165,248)(166,249)(333,416)(334,417)(335,418)(336,419)
(337,420)(338,421)(339,422)(340,423)(341,424)(342,425)(343,426)(344,427)
(345,428)(346,429)(347,430)(348,431)(349,432)(350,433)(351,434)(352,435)
(353,436)(354,437)(355,438)(356,439)(357,440)(358,441)(359,442)(360,443)
(361,444)(362,445)(363,446)(364,447)(365,448)(366,449)(367,450)(368,451)
(369,452)(370,453)(371,454)(372,455)(373,456)(374,457)(375,458)(376,459)
(377,460)(378,461)(379,462)(380,463)(381,464)(382,465)(383,466)(384,467)
(385,468)(386,469)(387,470)(388,471)(389,472)(390,473)(391,474)(392,475)
(393,476)(394,477)(395,478)(396,479)(397,480)(398,481)(399,482)(400,483)
(401,484)(402,485)(403,486)(404,487)(405,488)(406,489)(407,490)(408,491)
(409,492)(410,493)(411,494)(412,495)(413,496)(414,497)(415,498);
s1 := Sym(498)!( 1, 84)( 2,166)( 3,165)( 4,164)( 5,163)( 6,162)( 7,161)
( 8,160)( 9,159)( 10,158)( 11,157)( 12,156)( 13,155)( 14,154)( 15,153)
( 16,152)( 17,151)( 18,150)( 19,149)( 20,148)( 21,147)( 22,146)( 23,145)
( 24,144)( 25,143)( 26,142)( 27,141)( 28,140)( 29,139)( 30,138)( 31,137)
( 32,136)( 33,135)( 34,134)( 35,133)( 36,132)( 37,131)( 38,130)( 39,129)
( 40,128)( 41,127)( 42,126)( 43,125)( 44,124)( 45,123)( 46,122)( 47,121)
( 48,120)( 49,119)( 50,118)( 51,117)( 52,116)( 53,115)( 54,114)( 55,113)
( 56,112)( 57,111)( 58,110)( 59,109)( 60,108)( 61,107)( 62,106)( 63,105)
( 64,104)( 65,103)( 66,102)( 67,101)( 68,100)( 69, 99)( 70, 98)( 71, 97)
( 72, 96)( 73, 95)( 74, 94)( 75, 93)( 76, 92)( 77, 91)( 78, 90)( 79, 89)
( 80, 88)( 81, 87)( 82, 86)( 83, 85)(168,249)(169,248)(170,247)(171,246)
(172,245)(173,244)(174,243)(175,242)(176,241)(177,240)(178,239)(179,238)
(180,237)(181,236)(182,235)(183,234)(184,233)(185,232)(186,231)(187,230)
(188,229)(189,228)(190,227)(191,226)(192,225)(193,224)(194,223)(195,222)
(196,221)(197,220)(198,219)(199,218)(200,217)(201,216)(202,215)(203,214)
(204,213)(205,212)(206,211)(207,210)(208,209)(250,333)(251,415)(252,414)
(253,413)(254,412)(255,411)(256,410)(257,409)(258,408)(259,407)(260,406)
(261,405)(262,404)(263,403)(264,402)(265,401)(266,400)(267,399)(268,398)
(269,397)(270,396)(271,395)(272,394)(273,393)(274,392)(275,391)(276,390)
(277,389)(278,388)(279,387)(280,386)(281,385)(282,384)(283,383)(284,382)
(285,381)(286,380)(287,379)(288,378)(289,377)(290,376)(291,375)(292,374)
(293,373)(294,372)(295,371)(296,370)(297,369)(298,368)(299,367)(300,366)
(301,365)(302,364)(303,363)(304,362)(305,361)(306,360)(307,359)(308,358)
(309,357)(310,356)(311,355)(312,354)(313,353)(314,352)(315,351)(316,350)
(317,349)(318,348)(319,347)(320,346)(321,345)(322,344)(323,343)(324,342)
(325,341)(326,340)(327,339)(328,338)(329,337)(330,336)(331,335)(332,334)
(417,498)(418,497)(419,496)(420,495)(421,494)(422,493)(423,492)(424,491)
(425,490)(426,489)(427,488)(428,487)(429,486)(430,485)(431,484)(432,483)
(433,482)(434,481)(435,480)(436,479)(437,478)(438,477)(439,476)(440,475)
(441,474)(442,473)(443,472)(444,471)(445,470)(446,469)(447,468)(448,467)
(449,466)(450,465)(451,464)(452,463)(453,462)(454,461)(455,460)(456,459)
(457,458);
s2 := Sym(498)!( 1,251)( 2,250)( 3,332)( 4,331)( 5,330)( 6,329)( 7,328)
( 8,327)( 9,326)( 10,325)( 11,324)( 12,323)( 13,322)( 14,321)( 15,320)
( 16,319)( 17,318)( 18,317)( 19,316)( 20,315)( 21,314)( 22,313)( 23,312)
( 24,311)( 25,310)( 26,309)( 27,308)( 28,307)( 29,306)( 30,305)( 31,304)
( 32,303)( 33,302)( 34,301)( 35,300)( 36,299)( 37,298)( 38,297)( 39,296)
( 40,295)( 41,294)( 42,293)( 43,292)( 44,291)( 45,290)( 46,289)( 47,288)
( 48,287)( 49,286)( 50,285)( 51,284)( 52,283)( 53,282)( 54,281)( 55,280)
( 56,279)( 57,278)( 58,277)( 59,276)( 60,275)( 61,274)( 62,273)( 63,272)
( 64,271)( 65,270)( 66,269)( 67,268)( 68,267)( 69,266)( 70,265)( 71,264)
( 72,263)( 73,262)( 74,261)( 75,260)( 76,259)( 77,258)( 78,257)( 79,256)
( 80,255)( 81,254)( 82,253)( 83,252)( 84,334)( 85,333)( 86,415)( 87,414)
( 88,413)( 89,412)( 90,411)( 91,410)( 92,409)( 93,408)( 94,407)( 95,406)
( 96,405)( 97,404)( 98,403)( 99,402)(100,401)(101,400)(102,399)(103,398)
(104,397)(105,396)(106,395)(107,394)(108,393)(109,392)(110,391)(111,390)
(112,389)(113,388)(114,387)(115,386)(116,385)(117,384)(118,383)(119,382)
(120,381)(121,380)(122,379)(123,378)(124,377)(125,376)(126,375)(127,374)
(128,373)(129,372)(130,371)(131,370)(132,369)(133,368)(134,367)(135,366)
(136,365)(137,364)(138,363)(139,362)(140,361)(141,360)(142,359)(143,358)
(144,357)(145,356)(146,355)(147,354)(148,353)(149,352)(150,351)(151,350)
(152,349)(153,348)(154,347)(155,346)(156,345)(157,344)(158,343)(159,342)
(160,341)(161,340)(162,339)(163,338)(164,337)(165,336)(166,335)(167,417)
(168,416)(169,498)(170,497)(171,496)(172,495)(173,494)(174,493)(175,492)
(176,491)(177,490)(178,489)(179,488)(180,487)(181,486)(182,485)(183,484)
(184,483)(185,482)(186,481)(187,480)(188,479)(189,478)(190,477)(191,476)
(192,475)(193,474)(194,473)(195,472)(196,471)(197,470)(198,469)(199,468)
(200,467)(201,466)(202,465)(203,464)(204,463)(205,462)(206,461)(207,460)
(208,459)(209,458)(210,457)(211,456)(212,455)(213,454)(214,453)(215,452)
(216,451)(217,450)(218,449)(219,448)(220,447)(221,446)(222,445)(223,444)
(224,443)(225,442)(226,441)(227,440)(228,439)(229,438)(230,437)(231,436)
(232,435)(233,434)(234,433)(235,432)(236,431)(237,430)(238,429)(239,428)
(240,427)(241,426)(242,425)(243,424)(244,423)(245,422)(246,421)(247,420)
(248,419)(249,418);
poly := sub<Sym(498)|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*s2*s1*s0*s1*s2*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*s1*s2 >;
References : None.
to this polytope