Polytope of Type {6,164}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,164}*1968a
Also Known As : {6,164|2}. if this polytope has another name.
Group : SmallGroup(1968,133)
Rank : 3
Schlafli Type : {6,164}
Number of vertices, edges, etc : 6, 492, 164
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 : {6,82}*984
3-fold quotients : {2,164}*656
6-fold quotients : {2,82}*328
12-fold quotients : {2,41}*164
41-fold quotients : {6,4}*48a
82-fold quotients : {6,2}*24
123-fold quotients : {2,4}*16
164-fold quotients : {3,2}*12
246-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 42, 83)( 43, 84)( 44, 85)( 45, 86)( 46, 87)( 47, 88)( 48, 89)( 49, 90)
( 50, 91)( 51, 92)( 52, 93)( 53, 94)( 54, 95)( 55, 96)( 56, 97)( 57, 98)
( 58, 99)( 59,100)( 60,101)( 61,102)( 62,103)( 63,104)( 64,105)( 65,106)
( 66,107)( 67,108)( 68,109)( 69,110)( 70,111)( 71,112)( 72,113)( 73,114)
( 74,115)( 75,116)( 76,117)( 77,118)( 78,119)( 79,120)( 80,121)( 81,122)
( 82,123)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)(171,212)
(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)
(180,221)(181,222)(182,223)(183,224)(184,225)(185,226)(186,227)(187,228)
(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)
(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,244)
(204,245)(205,246)(288,329)(289,330)(290,331)(291,332)(292,333)(293,334)
(294,335)(295,336)(296,337)(297,338)(298,339)(299,340)(300,341)(301,342)
(302,343)(303,344)(304,345)(305,346)(306,347)(307,348)(308,349)(309,350)
(310,351)(311,352)(312,353)(313,354)(314,355)(315,356)(316,357)(317,358)
(318,359)(319,360)(320,361)(321,362)(322,363)(323,364)(324,365)(325,366)
(326,367)(327,368)(328,369)(411,452)(412,453)(413,454)(414,455)(415,456)
(416,457)(417,458)(418,459)(419,460)(420,461)(421,462)(422,463)(423,464)
(424,465)(425,466)(426,467)(427,468)(428,469)(429,470)(430,471)(431,472)
(432,473)(433,474)(434,475)(435,476)(436,477)(437,478)(438,479)(439,480)
(440,481)(441,482)(442,483)(443,484)(444,485)(445,486)(446,487)(447,488)
(448,489)(449,490)(450,491)(451,492);;
s1 := ( 1, 42)( 2, 82)( 3, 81)( 4, 80)( 5, 79)( 6, 78)( 7, 77)( 8, 76)
( 9, 75)( 10, 74)( 11, 73)( 12, 72)( 13, 71)( 14, 70)( 15, 69)( 16, 68)
( 17, 67)( 18, 66)( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)( 24, 60)
( 25, 59)( 26, 58)( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 53)( 32, 52)
( 33, 51)( 34, 50)( 35, 49)( 36, 48)( 37, 47)( 38, 46)( 39, 45)( 40, 44)
( 41, 43)( 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)(124,165)(125,205)(126,204)
(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,197)(134,196)
(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)(142,188)
(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180)
(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)(158,172)
(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(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)(247,411)(248,451)(249,450)(250,449)(251,448)(252,447)
(253,446)(254,445)(255,444)(256,443)(257,442)(258,441)(259,440)(260,439)
(261,438)(262,437)(263,436)(264,435)(265,434)(266,433)(267,432)(268,431)
(269,430)(270,429)(271,428)(272,427)(273,426)(274,425)(275,424)(276,423)
(277,422)(278,421)(279,420)(280,419)(281,418)(282,417)(283,416)(284,415)
(285,414)(286,413)(287,412)(288,370)(289,410)(290,409)(291,408)(292,407)
(293,406)(294,405)(295,404)(296,403)(297,402)(298,401)(299,400)(300,399)
(301,398)(302,397)(303,396)(304,395)(305,394)(306,393)(307,392)(308,391)
(309,390)(310,389)(311,388)(312,387)(313,386)(314,385)(315,384)(316,383)
(317,382)(318,381)(319,380)(320,379)(321,378)(322,377)(323,376)(324,375)
(325,374)(326,373)(327,372)(328,371)(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);;
s2 := ( 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)( 42,289)( 43,288)( 44,328)( 45,327)( 46,326)( 47,325)( 48,324)
( 49,323)( 50,322)( 51,321)( 52,320)( 53,319)( 54,318)( 55,317)( 56,316)
( 57,315)( 58,314)( 59,313)( 60,312)( 61,311)( 62,310)( 63,309)( 64,308)
( 65,307)( 66,306)( 67,305)( 68,304)( 69,303)( 70,302)( 71,301)( 72,300)
( 73,299)( 74,298)( 75,297)( 76,296)( 77,295)( 78,294)( 79,293)( 80,292)
( 81,291)( 82,290)( 83,330)( 84,329)( 85,369)( 86,368)( 87,367)( 88,366)
( 89,365)( 90,364)( 91,363)( 92,362)( 93,361)( 94,360)( 95,359)( 96,358)
( 97,357)( 98,356)( 99,355)(100,354)(101,353)(102,352)(103,351)(104,350)
(105,349)(106,348)(107,347)(108,346)(109,345)(110,344)(111,343)(112,342)
(113,341)(114,340)(115,339)(116,338)(117,337)(118,336)(119,335)(120,334)
(121,333)(122,332)(123,331)(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)(165,412)(166,411)(167,451)(168,450)
(169,449)(170,448)(171,447)(172,446)(173,445)(174,444)(175,443)(176,442)
(177,441)(178,440)(179,439)(180,438)(181,437)(182,436)(183,435)(184,434)
(185,433)(186,432)(187,431)(188,430)(189,429)(190,428)(191,427)(192,426)
(193,425)(194,424)(195,423)(196,422)(197,421)(198,420)(199,419)(200,418)
(201,417)(202,416)(203,415)(204,414)(205,413)(206,453)(207,452)(208,492)
(209,491)(210,490)(211,489)(212,488)(213,487)(214,486)(215,485)(216,484)
(217,483)(218,482)(219,481)(220,480)(221,479)(222,478)(223,477)(224,476)
(225,475)(226,474)(227,473)(228,472)(229,471)(230,470)(231,469)(232,468)
(233,467)(234,466)(235,465)(236,464)(237,463)(238,462)(239,461)(240,460)
(241,459)(242,458)(243,457)(244,456)(245,455)(246,454);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(492)!( 42, 83)( 43, 84)( 44, 85)( 45, 86)( 46, 87)( 47, 88)( 48, 89)
( 49, 90)( 50, 91)( 51, 92)( 52, 93)( 53, 94)( 54, 95)( 55, 96)( 56, 97)
( 57, 98)( 58, 99)( 59,100)( 60,101)( 61,102)( 62,103)( 63,104)( 64,105)
( 65,106)( 66,107)( 67,108)( 68,109)( 69,110)( 70,111)( 71,112)( 72,113)
( 73,114)( 74,115)( 75,116)( 76,117)( 77,118)( 78,119)( 79,120)( 80,121)
( 81,122)( 82,123)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)
(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)
(179,220)(180,221)(181,222)(182,223)(183,224)(184,225)(185,226)(186,227)
(187,228)(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)
(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)
(203,244)(204,245)(205,246)(288,329)(289,330)(290,331)(291,332)(292,333)
(293,334)(294,335)(295,336)(296,337)(297,338)(298,339)(299,340)(300,341)
(301,342)(302,343)(303,344)(304,345)(305,346)(306,347)(307,348)(308,349)
(309,350)(310,351)(311,352)(312,353)(313,354)(314,355)(315,356)(316,357)
(317,358)(318,359)(319,360)(320,361)(321,362)(322,363)(323,364)(324,365)
(325,366)(326,367)(327,368)(328,369)(411,452)(412,453)(413,454)(414,455)
(415,456)(416,457)(417,458)(418,459)(419,460)(420,461)(421,462)(422,463)
(423,464)(424,465)(425,466)(426,467)(427,468)(428,469)(429,470)(430,471)
(431,472)(432,473)(433,474)(434,475)(435,476)(436,477)(437,478)(438,479)
(439,480)(440,481)(441,482)(442,483)(443,484)(444,485)(445,486)(446,487)
(447,488)(448,489)(449,490)(450,491)(451,492);
s1 := Sym(492)!( 1, 42)( 2, 82)( 3, 81)( 4, 80)( 5, 79)( 6, 78)( 7, 77)
( 8, 76)( 9, 75)( 10, 74)( 11, 73)( 12, 72)( 13, 71)( 14, 70)( 15, 69)
( 16, 68)( 17, 67)( 18, 66)( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)
( 24, 60)( 25, 59)( 26, 58)( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 53)
( 32, 52)( 33, 51)( 34, 50)( 35, 49)( 36, 48)( 37, 47)( 38, 46)( 39, 45)
( 40, 44)( 41, 43)( 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)(124,165)(125,205)
(126,204)(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,197)
(134,196)(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)
(142,188)(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)
(150,180)(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)
(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(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)(247,411)(248,451)(249,450)(250,449)(251,448)
(252,447)(253,446)(254,445)(255,444)(256,443)(257,442)(258,441)(259,440)
(260,439)(261,438)(262,437)(263,436)(264,435)(265,434)(266,433)(267,432)
(268,431)(269,430)(270,429)(271,428)(272,427)(273,426)(274,425)(275,424)
(276,423)(277,422)(278,421)(279,420)(280,419)(281,418)(282,417)(283,416)
(284,415)(285,414)(286,413)(287,412)(288,370)(289,410)(290,409)(291,408)
(292,407)(293,406)(294,405)(295,404)(296,403)(297,402)(298,401)(299,400)
(300,399)(301,398)(302,397)(303,396)(304,395)(305,394)(306,393)(307,392)
(308,391)(309,390)(310,389)(311,388)(312,387)(313,386)(314,385)(315,384)
(316,383)(317,382)(318,381)(319,380)(320,379)(321,378)(322,377)(323,376)
(324,375)(325,374)(326,373)(327,372)(328,371)(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);
s2 := 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)( 42,289)( 43,288)( 44,328)( 45,327)( 46,326)( 47,325)
( 48,324)( 49,323)( 50,322)( 51,321)( 52,320)( 53,319)( 54,318)( 55,317)
( 56,316)( 57,315)( 58,314)( 59,313)( 60,312)( 61,311)( 62,310)( 63,309)
( 64,308)( 65,307)( 66,306)( 67,305)( 68,304)( 69,303)( 70,302)( 71,301)
( 72,300)( 73,299)( 74,298)( 75,297)( 76,296)( 77,295)( 78,294)( 79,293)
( 80,292)( 81,291)( 82,290)( 83,330)( 84,329)( 85,369)( 86,368)( 87,367)
( 88,366)( 89,365)( 90,364)( 91,363)( 92,362)( 93,361)( 94,360)( 95,359)
( 96,358)( 97,357)( 98,356)( 99,355)(100,354)(101,353)(102,352)(103,351)
(104,350)(105,349)(106,348)(107,347)(108,346)(109,345)(110,344)(111,343)
(112,342)(113,341)(114,340)(115,339)(116,338)(117,337)(118,336)(119,335)
(120,334)(121,333)(122,332)(123,331)(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)(165,412)(166,411)(167,451)
(168,450)(169,449)(170,448)(171,447)(172,446)(173,445)(174,444)(175,443)
(176,442)(177,441)(178,440)(179,439)(180,438)(181,437)(182,436)(183,435)
(184,434)(185,433)(186,432)(187,431)(188,430)(189,429)(190,428)(191,427)
(192,426)(193,425)(194,424)(195,423)(196,422)(197,421)(198,420)(199,419)
(200,418)(201,417)(202,416)(203,415)(204,414)(205,413)(206,453)(207,452)
(208,492)(209,491)(210,490)(211,489)(212,488)(213,487)(214,486)(215,485)
(216,484)(217,483)(218,482)(219,481)(220,480)(221,479)(222,478)(223,477)
(224,476)(225,475)(226,474)(227,473)(228,472)(229,471)(230,470)(231,469)
(232,468)(233,467)(234,466)(235,465)(236,464)(237,463)(238,462)(239,461)
(240,460)(241,459)(242,458)(243,457)(244,456)(245,455)(246,454);
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,
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 >;
References : None.
to this polytope