Polytope of Type {4,234}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,234}*1872a
Also Known As : {4,234|2}. if this polytope has another name.
Group : SmallGroup(1872,238)
Rank : 3
Schlafli Type : {4,234}
Number of vertices, edges, etc : 4, 468, 234
Order of s0s1s2 : 468
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 : {2,234}*936
3-fold quotients : {4,78}*624a
4-fold quotients : {2,117}*468
6-fold quotients : {2,78}*312
9-fold quotients : {4,26}*208
12-fold quotients : {2,39}*156
13-fold quotients : {4,18}*144a
18-fold quotients : {2,26}*104
26-fold quotients : {2,18}*72
36-fold quotients : {2,13}*52
39-fold quotients : {4,6}*48a
52-fold quotients : {2,9}*36
78-fold quotients : {2,6}*24
117-fold quotients : {4,2}*16
156-fold quotients : {2,3}*12
234-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (235,352)(236,353)(237,354)(238,355)(239,356)(240,357)(241,358)(242,359)
(243,360)(244,361)(245,362)(246,363)(247,364)(248,365)(249,366)(250,367)
(251,368)(252,369)(253,370)(254,371)(255,372)(256,373)(257,374)(258,375)
(259,376)(260,377)(261,378)(262,379)(263,380)(264,381)(265,382)(266,383)
(267,384)(268,385)(269,386)(270,387)(271,388)(272,389)(273,390)(274,391)
(275,392)(276,393)(277,394)(278,395)(279,396)(280,397)(281,398)(282,399)
(283,400)(284,401)(285,402)(286,403)(287,404)(288,405)(289,406)(290,407)
(291,408)(292,409)(293,410)(294,411)(295,412)(296,413)(297,414)(298,415)
(299,416)(300,417)(301,418)(302,419)(303,420)(304,421)(305,422)(306,423)
(307,424)(308,425)(309,426)(310,427)(311,428)(312,429)(313,430)(314,431)
(315,432)(316,433)(317,434)(318,435)(319,436)(320,437)(321,438)(322,439)
(323,440)(324,441)(325,442)(326,443)(327,444)(328,445)(329,446)(330,447)
(331,448)(332,449)(333,450)(334,451)(335,452)(336,453)(337,454)(338,455)
(339,456)(340,457)(341,458)(342,459)(343,460)(344,461)(345,462)(346,463)
(347,464)(348,465)(349,466)(350,467)(351,468);;
s1 := ( 1,235)( 2,237)( 3,236)( 4,271)( 5,273)( 6,272)( 7,268)( 8,270)
( 9,269)( 10,265)( 11,267)( 12,266)( 13,262)( 14,264)( 15,263)( 16,259)
( 17,261)( 18,260)( 19,256)( 20,258)( 21,257)( 22,253)( 23,255)( 24,254)
( 25,250)( 26,252)( 27,251)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,241)( 35,243)( 36,242)( 37,238)( 38,240)( 39,239)( 40,315)
( 41,314)( 42,313)( 43,351)( 44,350)( 45,349)( 46,348)( 47,347)( 48,346)
( 49,345)( 50,344)( 51,343)( 52,342)( 53,341)( 54,340)( 55,339)( 56,338)
( 57,337)( 58,336)( 59,335)( 60,334)( 61,333)( 62,332)( 63,331)( 64,330)
( 65,329)( 66,328)( 67,327)( 68,326)( 69,325)( 70,324)( 71,323)( 72,322)
( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,276)( 80,275)
( 81,274)( 82,312)( 83,311)( 84,310)( 85,309)( 86,308)( 87,307)( 88,306)
( 89,305)( 90,304)( 91,303)( 92,302)( 93,301)( 94,300)( 95,299)( 96,298)
( 97,297)( 98,296)( 99,295)(100,294)(101,293)(102,292)(103,291)(104,290)
(105,289)(106,288)(107,287)(108,286)(109,285)(110,284)(111,283)(112,282)
(113,281)(114,280)(115,279)(116,278)(117,277)(118,352)(119,354)(120,353)
(121,388)(122,390)(123,389)(124,385)(125,387)(126,386)(127,382)(128,384)
(129,383)(130,379)(131,381)(132,380)(133,376)(134,378)(135,377)(136,373)
(137,375)(138,374)(139,370)(140,372)(141,371)(142,367)(143,369)(144,368)
(145,364)(146,366)(147,365)(148,361)(149,363)(150,362)(151,358)(152,360)
(153,359)(154,355)(155,357)(156,356)(157,432)(158,431)(159,430)(160,468)
(161,467)(162,466)(163,465)(164,464)(165,463)(166,462)(167,461)(168,460)
(169,459)(170,458)(171,457)(172,456)(173,455)(174,454)(175,453)(176,452)
(177,451)(178,450)(179,449)(180,448)(181,447)(182,446)(183,445)(184,444)
(185,443)(186,442)(187,441)(188,440)(189,439)(190,438)(191,437)(192,436)
(193,435)(194,434)(195,433)(196,393)(197,392)(198,391)(199,429)(200,428)
(201,427)(202,426)(203,425)(204,424)(205,423)(206,422)(207,421)(208,420)
(209,419)(210,418)(211,417)(212,416)(213,415)(214,414)(215,413)(216,412)
(217,411)(218,410)(219,409)(220,408)(221,407)(222,406)(223,405)(224,404)
(225,403)(226,402)(227,401)(228,400)(229,399)(230,398)(231,397)(232,396)
(233,395)(234,394);;
s2 := ( 1, 43)( 2, 45)( 3, 44)( 4, 40)( 5, 42)( 6, 41)( 7, 76)( 8, 78)
( 9, 77)( 10, 73)( 11, 75)( 12, 74)( 13, 70)( 14, 72)( 15, 71)( 16, 67)
( 17, 69)( 18, 68)( 19, 64)( 20, 66)( 21, 65)( 22, 61)( 23, 63)( 24, 62)
( 25, 58)( 26, 60)( 27, 59)( 28, 55)( 29, 57)( 30, 56)( 31, 52)( 32, 54)
( 33, 53)( 34, 49)( 35, 51)( 36, 50)( 37, 46)( 38, 48)( 39, 47)( 79, 84)
( 80, 83)( 81, 82)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)( 90,112)
( 91,111)( 92,110)( 93,109)( 94,108)( 95,107)( 96,106)( 97,105)( 98,104)
( 99,103)(100,102)(118,160)(119,162)(120,161)(121,157)(122,159)(123,158)
(124,193)(125,195)(126,194)(127,190)(128,192)(129,191)(130,187)(131,189)
(132,188)(133,184)(134,186)(135,185)(136,181)(137,183)(138,182)(139,178)
(140,180)(141,179)(142,175)(143,177)(144,176)(145,172)(146,174)(147,173)
(148,169)(149,171)(150,170)(151,166)(152,168)(153,167)(154,163)(155,165)
(156,164)(196,201)(197,200)(198,199)(202,234)(203,233)(204,232)(205,231)
(206,230)(207,229)(208,228)(209,227)(210,226)(211,225)(212,224)(213,223)
(214,222)(215,221)(216,220)(217,219)(235,277)(236,279)(237,278)(238,274)
(239,276)(240,275)(241,310)(242,312)(243,311)(244,307)(245,309)(246,308)
(247,304)(248,306)(249,305)(250,301)(251,303)(252,302)(253,298)(254,300)
(255,299)(256,295)(257,297)(258,296)(259,292)(260,294)(261,293)(262,289)
(263,291)(264,290)(265,286)(266,288)(267,287)(268,283)(269,285)(270,284)
(271,280)(272,282)(273,281)(313,318)(314,317)(315,316)(319,351)(320,350)
(321,349)(322,348)(323,347)(324,346)(325,345)(326,344)(327,343)(328,342)
(329,341)(330,340)(331,339)(332,338)(333,337)(334,336)(352,394)(353,396)
(354,395)(355,391)(356,393)(357,392)(358,427)(359,429)(360,428)(361,424)
(362,426)(363,425)(364,421)(365,423)(366,422)(367,418)(368,420)(369,419)
(370,415)(371,417)(372,416)(373,412)(374,414)(375,413)(376,409)(377,411)
(378,410)(379,406)(380,408)(381,407)(382,403)(383,405)(384,404)(385,400)
(386,402)(387,401)(388,397)(389,399)(390,398)(430,435)(431,434)(432,433)
(436,468)(437,467)(438,466)(439,465)(440,464)(441,463)(442,462)(443,461)
(444,460)(445,459)(446,458)(447,457)(448,456)(449,455)(450,454)(451,453);;
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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(468)!(235,352)(236,353)(237,354)(238,355)(239,356)(240,357)(241,358)
(242,359)(243,360)(244,361)(245,362)(246,363)(247,364)(248,365)(249,366)
(250,367)(251,368)(252,369)(253,370)(254,371)(255,372)(256,373)(257,374)
(258,375)(259,376)(260,377)(261,378)(262,379)(263,380)(264,381)(265,382)
(266,383)(267,384)(268,385)(269,386)(270,387)(271,388)(272,389)(273,390)
(274,391)(275,392)(276,393)(277,394)(278,395)(279,396)(280,397)(281,398)
(282,399)(283,400)(284,401)(285,402)(286,403)(287,404)(288,405)(289,406)
(290,407)(291,408)(292,409)(293,410)(294,411)(295,412)(296,413)(297,414)
(298,415)(299,416)(300,417)(301,418)(302,419)(303,420)(304,421)(305,422)
(306,423)(307,424)(308,425)(309,426)(310,427)(311,428)(312,429)(313,430)
(314,431)(315,432)(316,433)(317,434)(318,435)(319,436)(320,437)(321,438)
(322,439)(323,440)(324,441)(325,442)(326,443)(327,444)(328,445)(329,446)
(330,447)(331,448)(332,449)(333,450)(334,451)(335,452)(336,453)(337,454)
(338,455)(339,456)(340,457)(341,458)(342,459)(343,460)(344,461)(345,462)
(346,463)(347,464)(348,465)(349,466)(350,467)(351,468);
s1 := Sym(468)!( 1,235)( 2,237)( 3,236)( 4,271)( 5,273)( 6,272)( 7,268)
( 8,270)( 9,269)( 10,265)( 11,267)( 12,266)( 13,262)( 14,264)( 15,263)
( 16,259)( 17,261)( 18,260)( 19,256)( 20,258)( 21,257)( 22,253)( 23,255)
( 24,254)( 25,250)( 26,252)( 27,251)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,241)( 35,243)( 36,242)( 37,238)( 38,240)( 39,239)
( 40,315)( 41,314)( 42,313)( 43,351)( 44,350)( 45,349)( 46,348)( 47,347)
( 48,346)( 49,345)( 50,344)( 51,343)( 52,342)( 53,341)( 54,340)( 55,339)
( 56,338)( 57,337)( 58,336)( 59,335)( 60,334)( 61,333)( 62,332)( 63,331)
( 64,330)( 65,329)( 66,328)( 67,327)( 68,326)( 69,325)( 70,324)( 71,323)
( 72,322)( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,276)
( 80,275)( 81,274)( 82,312)( 83,311)( 84,310)( 85,309)( 86,308)( 87,307)
( 88,306)( 89,305)( 90,304)( 91,303)( 92,302)( 93,301)( 94,300)( 95,299)
( 96,298)( 97,297)( 98,296)( 99,295)(100,294)(101,293)(102,292)(103,291)
(104,290)(105,289)(106,288)(107,287)(108,286)(109,285)(110,284)(111,283)
(112,282)(113,281)(114,280)(115,279)(116,278)(117,277)(118,352)(119,354)
(120,353)(121,388)(122,390)(123,389)(124,385)(125,387)(126,386)(127,382)
(128,384)(129,383)(130,379)(131,381)(132,380)(133,376)(134,378)(135,377)
(136,373)(137,375)(138,374)(139,370)(140,372)(141,371)(142,367)(143,369)
(144,368)(145,364)(146,366)(147,365)(148,361)(149,363)(150,362)(151,358)
(152,360)(153,359)(154,355)(155,357)(156,356)(157,432)(158,431)(159,430)
(160,468)(161,467)(162,466)(163,465)(164,464)(165,463)(166,462)(167,461)
(168,460)(169,459)(170,458)(171,457)(172,456)(173,455)(174,454)(175,453)
(176,452)(177,451)(178,450)(179,449)(180,448)(181,447)(182,446)(183,445)
(184,444)(185,443)(186,442)(187,441)(188,440)(189,439)(190,438)(191,437)
(192,436)(193,435)(194,434)(195,433)(196,393)(197,392)(198,391)(199,429)
(200,428)(201,427)(202,426)(203,425)(204,424)(205,423)(206,422)(207,421)
(208,420)(209,419)(210,418)(211,417)(212,416)(213,415)(214,414)(215,413)
(216,412)(217,411)(218,410)(219,409)(220,408)(221,407)(222,406)(223,405)
(224,404)(225,403)(226,402)(227,401)(228,400)(229,399)(230,398)(231,397)
(232,396)(233,395)(234,394);
s2 := Sym(468)!( 1, 43)( 2, 45)( 3, 44)( 4, 40)( 5, 42)( 6, 41)( 7, 76)
( 8, 78)( 9, 77)( 10, 73)( 11, 75)( 12, 74)( 13, 70)( 14, 72)( 15, 71)
( 16, 67)( 17, 69)( 18, 68)( 19, 64)( 20, 66)( 21, 65)( 22, 61)( 23, 63)
( 24, 62)( 25, 58)( 26, 60)( 27, 59)( 28, 55)( 29, 57)( 30, 56)( 31, 52)
( 32, 54)( 33, 53)( 34, 49)( 35, 51)( 36, 50)( 37, 46)( 38, 48)( 39, 47)
( 79, 84)( 80, 83)( 81, 82)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)
( 90,112)( 91,111)( 92,110)( 93,109)( 94,108)( 95,107)( 96,106)( 97,105)
( 98,104)( 99,103)(100,102)(118,160)(119,162)(120,161)(121,157)(122,159)
(123,158)(124,193)(125,195)(126,194)(127,190)(128,192)(129,191)(130,187)
(131,189)(132,188)(133,184)(134,186)(135,185)(136,181)(137,183)(138,182)
(139,178)(140,180)(141,179)(142,175)(143,177)(144,176)(145,172)(146,174)
(147,173)(148,169)(149,171)(150,170)(151,166)(152,168)(153,167)(154,163)
(155,165)(156,164)(196,201)(197,200)(198,199)(202,234)(203,233)(204,232)
(205,231)(206,230)(207,229)(208,228)(209,227)(210,226)(211,225)(212,224)
(213,223)(214,222)(215,221)(216,220)(217,219)(235,277)(236,279)(237,278)
(238,274)(239,276)(240,275)(241,310)(242,312)(243,311)(244,307)(245,309)
(246,308)(247,304)(248,306)(249,305)(250,301)(251,303)(252,302)(253,298)
(254,300)(255,299)(256,295)(257,297)(258,296)(259,292)(260,294)(261,293)
(262,289)(263,291)(264,290)(265,286)(266,288)(267,287)(268,283)(269,285)
(270,284)(271,280)(272,282)(273,281)(313,318)(314,317)(315,316)(319,351)
(320,350)(321,349)(322,348)(323,347)(324,346)(325,345)(326,344)(327,343)
(328,342)(329,341)(330,340)(331,339)(332,338)(333,337)(334,336)(352,394)
(353,396)(354,395)(355,391)(356,393)(357,392)(358,427)(359,429)(360,428)
(361,424)(362,426)(363,425)(364,421)(365,423)(366,422)(367,418)(368,420)
(369,419)(370,415)(371,417)(372,416)(373,412)(374,414)(375,413)(376,409)
(377,411)(378,410)(379,406)(380,408)(381,407)(382,403)(383,405)(384,404)
(385,400)(386,402)(387,401)(388,397)(389,399)(390,398)(430,435)(431,434)
(432,433)(436,468)(437,467)(438,466)(439,465)(440,464)(441,463)(442,462)
(443,461)(444,460)(445,459)(446,458)(447,457)(448,456)(449,455)(450,454)
(451,453);
poly := sub<Sym(468)|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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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