Polytope of Type {164,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {164,6}*1968a
Also Known As : {164,6|2}. if this polytope has another name.
Group : SmallGroup(1968,133)
Rank : 3
Schlafli Type : {164,6}
Number of vertices, edges, etc : 164, 492, 6
Order of s₀s₁s₂ : 492
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) :
   2-fold quotients : {82,6}*984
   3-fold quotients : {164,2}*656
   6-fold quotients : {82,2}*328
   12-fold quotients : {41,2}*164
   41-fold quotients : {4,6}*48a
   82-fold quotients : {2,6}*24
   123-fold quotients : {4,2}*16
   164-fold quotients : {2,3}*12
   246-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 41)(  3, 40)(  4, 39)(  5, 38)(  6, 37)(  7, 36)(  8, 35)(  9, 34)
( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)( 17, 26)
( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 43, 82)( 44, 81)( 45, 80)( 46, 79)
( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)( 54, 71)
( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)( 62, 63)
( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)( 91,116)
( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)( 99,108)
(100,107)(101,106)(102,105)(103,104)(125,164)(126,163)(127,162)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)
(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)
(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)
(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)
(182,189)(183,188)(184,187)(185,186)(207,246)(208,245)(209,244)(210,243)
(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)(217,236)(218,235)
(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)(225,228)(226,227)
(247,370)(248,410)(249,409)(250,408)(251,407)(252,406)(253,405)(254,404)
(255,403)(256,402)(257,401)(258,400)(259,399)(260,398)(261,397)(262,396)
(263,395)(264,394)(265,393)(266,392)(267,391)(268,390)(269,389)(270,388)
(271,387)(272,386)(273,385)(274,384)(275,383)(276,382)(277,381)(278,380)
(279,379)(280,378)(281,377)(282,376)(283,375)(284,374)(285,373)(286,372)
(287,371)(288,411)(289,451)(290,450)(291,449)(292,448)(293,447)(294,446)
(295,445)(296,444)(297,443)(298,442)(299,441)(300,440)(301,439)(302,438)
(303,437)(304,436)(305,435)(306,434)(307,433)(308,432)(309,431)(310,430)
(311,429)(312,428)(313,427)(314,426)(315,425)(316,424)(317,423)(318,422)
(319,421)(320,420)(321,419)(322,418)(323,417)(324,416)(325,415)(326,414)
(327,413)(328,412)(329,452)(330,492)(331,491)(332,490)(333,489)(334,488)
(335,487)(336,486)(337,485)(338,484)(339,483)(340,482)(341,481)(342,480)
(343,479)(344,478)(345,477)(346,476)(347,475)(348,474)(349,473)(350,472)
(351,471)(352,470)(353,469)(354,468)(355,467)(356,466)(357,465)(358,464)
(359,463)(360,462)(361,461)(362,460)(363,459)(364,458)(365,457)(366,456)
(367,455)(368,454)(369,453);;
s1 := (  1,248)(  2,247)(  3,287)(  4,286)(  5,285)(  6,284)(  7,283)(  8,282)
(  9,281)( 10,280)( 11,279)( 12,278)( 13,277)( 14,276)( 15,275)( 16,274)
( 17,273)( 18,272)( 19,271)( 20,270)( 21,269)( 22,268)( 23,267)( 24,266)
( 25,265)( 26,264)( 27,263)( 28,262)( 29,261)( 30,260)( 31,259)( 32,258)
( 33,257)( 34,256)( 35,255)( 36,254)( 37,253)( 38,252)( 39,251)( 40,250)
( 41,249)( 42,330)( 43,329)( 44,369)( 45,368)( 46,367)( 47,366)( 48,365)
( 49,364)( 50,363)( 51,362)( 52,361)( 53,360)( 54,359)( 55,358)( 56,357)
( 57,356)( 58,355)( 59,354)( 60,353)( 61,352)( 62,351)( 63,350)( 64,349)
( 65,348)( 66,347)( 67,346)( 68,345)( 69,344)( 70,343)( 71,342)( 72,341)
( 73,340)( 74,339)( 75,338)( 76,337)( 77,336)( 78,335)( 79,334)( 80,333)
( 81,332)( 82,331)( 83,289)( 84,288)( 85,328)( 86,327)( 87,326)( 88,325)
( 89,324)( 90,323)( 91,322)( 92,321)( 93,320)( 94,319)( 95,318)( 96,317)
( 97,316)( 98,315)( 99,314)(100,313)(101,312)(102,311)(103,310)(104,309)
(105,308)(106,307)(107,306)(108,305)(109,304)(110,303)(111,302)(112,301)
(113,300)(114,299)(115,298)(116,297)(117,296)(118,295)(119,294)(120,293)
(121,292)(122,291)(123,290)(124,371)(125,370)(126,410)(127,409)(128,408)
(129,407)(130,406)(131,405)(132,404)(133,403)(134,402)(135,401)(136,400)
(137,399)(138,398)(139,397)(140,396)(141,395)(142,394)(143,393)(144,392)
(145,391)(146,390)(147,389)(148,388)(149,387)(150,386)(151,385)(152,384)
(153,383)(154,382)(155,381)(156,380)(157,379)(158,378)(159,377)(160,376)
(161,375)(162,374)(163,373)(164,372)(165,453)(166,452)(167,492)(168,491)
(169,490)(170,489)(171,488)(172,487)(173,486)(174,485)(175,484)(176,483)
(177,482)(178,481)(179,480)(180,479)(181,478)(182,477)(183,476)(184,475)
(185,474)(186,473)(187,472)(188,471)(189,470)(190,469)(191,468)(192,467)
(193,466)(194,465)(195,464)(196,463)(197,462)(198,461)(199,460)(200,459)
(201,458)(202,457)(203,456)(204,455)(205,454)(206,412)(207,411)(208,451)
(209,450)(210,449)(211,448)(212,447)(213,446)(214,445)(215,444)(216,443)
(217,442)(218,441)(219,440)(220,439)(221,438)(222,437)(223,436)(224,435)
(225,434)(226,433)(227,432)(228,431)(229,430)(230,429)(231,428)(232,427)
(233,426)(234,425)(235,424)(236,423)(237,422)(238,421)(239,420)(240,419)
(241,418)(242,417)(243,416)(244,415)(245,414)(246,413);;
s2 := (  1, 42)(  2, 43)(  3, 44)(  4, 45)(  5, 46)(  6, 47)(  7, 48)(  8, 49)
(  9, 50)( 10, 51)( 11, 52)( 12, 53)( 13, 54)( 14, 55)( 15, 56)( 16, 57)
( 17, 58)( 18, 59)( 19, 60)( 20, 61)( 21, 62)( 22, 63)( 23, 64)( 24, 65)
( 25, 66)( 26, 67)( 27, 68)( 28, 69)( 29, 70)( 30, 71)( 31, 72)( 32, 73)
( 33, 74)( 34, 75)( 35, 76)( 36, 77)( 37, 78)( 38, 79)( 39, 80)( 40, 81)
( 41, 82)(124,165)(125,166)(126,167)(127,168)(128,169)(129,170)(130,171)
(131,172)(132,173)(133,174)(134,175)(135,176)(136,177)(137,178)(138,179)
(139,180)(140,181)(141,182)(142,183)(143,184)(144,185)(145,186)(146,187)
(147,188)(148,189)(149,190)(150,191)(151,192)(152,193)(153,194)(154,195)
(155,196)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)(162,203)
(163,204)(164,205)(247,288)(248,289)(249,290)(250,291)(251,292)(252,293)
(253,294)(254,295)(255,296)(256,297)(257,298)(258,299)(259,300)(260,301)
(261,302)(262,303)(263,304)(264,305)(265,306)(266,307)(267,308)(268,309)
(269,310)(270,311)(271,312)(272,313)(273,314)(274,315)(275,316)(276,317)
(277,318)(278,319)(279,320)(280,321)(281,322)(282,323)(283,324)(284,325)
(285,326)(286,327)(287,328)(370,411)(371,412)(372,413)(373,414)(374,415)
(375,416)(376,417)(377,418)(378,419)(379,420)(380,421)(381,422)(382,423)
(383,424)(384,425)(385,426)(386,427)(387,428)(388,429)(389,430)(390,431)
(391,432)(392,433)(393,434)(394,435)(395,436)(396,437)(397,438)(398,439)
(399,440)(400,441)(401,442)(402,443)(403,444)(404,445)(405,446)(406,447)
(407,448)(408,449)(409,450)(410,451);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(492)!(  2, 41)(  3, 40)(  4, 39)(  5, 38)(  6, 37)(  7, 36)(  8, 35)
(  9, 34)( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)
( 17, 26)( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 43, 82)( 44, 81)( 45, 80)
( 46, 79)( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)
( 54, 71)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)
( 62, 63)( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)
( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)
( 99,108)(100,107)(101,106)(102,105)(103,104)(125,164)(126,163)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)
(144,145)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)
(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)
(181,190)(182,189)(183,188)(184,187)(185,186)(207,246)(208,245)(209,244)
(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)(217,236)
(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)(225,228)
(226,227)(247,370)(248,410)(249,409)(250,408)(251,407)(252,406)(253,405)
(254,404)(255,403)(256,402)(257,401)(258,400)(259,399)(260,398)(261,397)
(262,396)(263,395)(264,394)(265,393)(266,392)(267,391)(268,390)(269,389)
(270,388)(271,387)(272,386)(273,385)(274,384)(275,383)(276,382)(277,381)
(278,380)(279,379)(280,378)(281,377)(282,376)(283,375)(284,374)(285,373)
(286,372)(287,371)(288,411)(289,451)(290,450)(291,449)(292,448)(293,447)
(294,446)(295,445)(296,444)(297,443)(298,442)(299,441)(300,440)(301,439)
(302,438)(303,437)(304,436)(305,435)(306,434)(307,433)(308,432)(309,431)
(310,430)(311,429)(312,428)(313,427)(314,426)(315,425)(316,424)(317,423)
(318,422)(319,421)(320,420)(321,419)(322,418)(323,417)(324,416)(325,415)
(326,414)(327,413)(328,412)(329,452)(330,492)(331,491)(332,490)(333,489)
(334,488)(335,487)(336,486)(337,485)(338,484)(339,483)(340,482)(341,481)
(342,480)(343,479)(344,478)(345,477)(346,476)(347,475)(348,474)(349,473)
(350,472)(351,471)(352,470)(353,469)(354,468)(355,467)(356,466)(357,465)
(358,464)(359,463)(360,462)(361,461)(362,460)(363,459)(364,458)(365,457)
(366,456)(367,455)(368,454)(369,453);
s1 := Sym(492)!(  1,248)(  2,247)(  3,287)(  4,286)(  5,285)(  6,284)(  7,283)
(  8,282)(  9,281)( 10,280)( 11,279)( 12,278)( 13,277)( 14,276)( 15,275)
( 16,274)( 17,273)( 18,272)( 19,271)( 20,270)( 21,269)( 22,268)( 23,267)
( 24,266)( 25,265)( 26,264)( 27,263)( 28,262)( 29,261)( 30,260)( 31,259)
( 32,258)( 33,257)( 34,256)( 35,255)( 36,254)( 37,253)( 38,252)( 39,251)
( 40,250)( 41,249)( 42,330)( 43,329)( 44,369)( 45,368)( 46,367)( 47,366)
( 48,365)( 49,364)( 50,363)( 51,362)( 52,361)( 53,360)( 54,359)( 55,358)
( 56,357)( 57,356)( 58,355)( 59,354)( 60,353)( 61,352)( 62,351)( 63,350)
( 64,349)( 65,348)( 66,347)( 67,346)( 68,345)( 69,344)( 70,343)( 71,342)
( 72,341)( 73,340)( 74,339)( 75,338)( 76,337)( 77,336)( 78,335)( 79,334)
( 80,333)( 81,332)( 82,331)( 83,289)( 84,288)( 85,328)( 86,327)( 87,326)
( 88,325)( 89,324)( 90,323)( 91,322)( 92,321)( 93,320)( 94,319)( 95,318)
( 96,317)( 97,316)( 98,315)( 99,314)(100,313)(101,312)(102,311)(103,310)
(104,309)(105,308)(106,307)(107,306)(108,305)(109,304)(110,303)(111,302)
(112,301)(113,300)(114,299)(115,298)(116,297)(117,296)(118,295)(119,294)
(120,293)(121,292)(122,291)(123,290)(124,371)(125,370)(126,410)(127,409)
(128,408)(129,407)(130,406)(131,405)(132,404)(133,403)(134,402)(135,401)
(136,400)(137,399)(138,398)(139,397)(140,396)(141,395)(142,394)(143,393)
(144,392)(145,391)(146,390)(147,389)(148,388)(149,387)(150,386)(151,385)
(152,384)(153,383)(154,382)(155,381)(156,380)(157,379)(158,378)(159,377)
(160,376)(161,375)(162,374)(163,373)(164,372)(165,453)(166,452)(167,492)
(168,491)(169,490)(170,489)(171,488)(172,487)(173,486)(174,485)(175,484)
(176,483)(177,482)(178,481)(179,480)(180,479)(181,478)(182,477)(183,476)
(184,475)(185,474)(186,473)(187,472)(188,471)(189,470)(190,469)(191,468)
(192,467)(193,466)(194,465)(195,464)(196,463)(197,462)(198,461)(199,460)
(200,459)(201,458)(202,457)(203,456)(204,455)(205,454)(206,412)(207,411)
(208,451)(209,450)(210,449)(211,448)(212,447)(213,446)(214,445)(215,444)
(216,443)(217,442)(218,441)(219,440)(220,439)(221,438)(222,437)(223,436)
(224,435)(225,434)(226,433)(227,432)(228,431)(229,430)(230,429)(231,428)
(232,427)(233,426)(234,425)(235,424)(236,423)(237,422)(238,421)(239,420)
(240,419)(241,418)(242,417)(243,416)(244,415)(245,414)(246,413);
s2 := Sym(492)!(  1, 42)(  2, 43)(  3, 44)(  4, 45)(  5, 46)(  6, 47)(  7, 48)
(  8, 49)(  9, 50)( 10, 51)( 11, 52)( 12, 53)( 13, 54)( 14, 55)( 15, 56)
( 16, 57)( 17, 58)( 18, 59)( 19, 60)( 20, 61)( 21, 62)( 22, 63)( 23, 64)
( 24, 65)( 25, 66)( 26, 67)( 27, 68)( 28, 69)( 29, 70)( 30, 71)( 31, 72)
( 32, 73)( 33, 74)( 34, 75)( 35, 76)( 36, 77)( 37, 78)( 38, 79)( 39, 80)
( 40, 81)( 41, 82)(124,165)(125,166)(126,167)(127,168)(128,169)(129,170)
(130,171)(131,172)(132,173)(133,174)(134,175)(135,176)(136,177)(137,178)
(138,179)(139,180)(140,181)(141,182)(142,183)(143,184)(144,185)(145,186)
(146,187)(147,188)(148,189)(149,190)(150,191)(151,192)(152,193)(153,194)
(154,195)(155,196)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)
(162,203)(163,204)(164,205)(247,288)(248,289)(249,290)(250,291)(251,292)
(252,293)(253,294)(254,295)(255,296)(256,297)(257,298)(258,299)(259,300)
(260,301)(261,302)(262,303)(263,304)(264,305)(265,306)(266,307)(267,308)
(268,309)(269,310)(270,311)(271,312)(272,313)(273,314)(274,315)(275,316)
(276,317)(277,318)(278,319)(279,320)(280,321)(281,322)(282,323)(283,324)
(284,325)(285,326)(286,327)(287,328)(370,411)(371,412)(372,413)(373,414)
(374,415)(375,416)(376,417)(377,418)(378,419)(379,420)(380,421)(381,422)
(382,423)(383,424)(384,425)(385,426)(386,427)(387,428)(388,429)(389,430)
(390,431)(391,432)(392,433)(393,434)(394,435)(395,436)(396,437)(397,438)
(398,439)(399,440)(400,441)(401,442)(402,443)(403,444)(404,445)(405,446)
(406,447)(407,448)(408,449)(409,450)(410,451);
poly := sub<Sym(492)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope