Polytope of Type {4,228}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,228}*1824a
Also Known As : {4,228|2}. if this polytope has another name.
Group : SmallGroup(1824,948)
Rank : 3
Schlafli Type : {4,228}
Number of vertices, edges, etc : 4, 456, 228
Order of s0s1s2 : 228
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 : {2,228}*912, {4,114}*912a
3-fold quotients : {4,76}*608
4-fold quotients : {2,114}*456
6-fold quotients : {2,76}*304, {4,38}*304
8-fold quotients : {2,57}*228
12-fold quotients : {2,38}*152
19-fold quotients : {4,12}*96a
24-fold quotients : {2,19}*76
38-fold quotients : {2,12}*48, {4,6}*48a
57-fold quotients : {4,4}*32
76-fold quotients : {2,6}*24
114-fold quotients : {2,4}*16, {4,2}*16
152-fold quotients : {2,3}*12
228-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (229,343)(230,344)(231,345)(232,346)(233,347)(234,348)(235,349)(236,350)
(237,351)(238,352)(239,353)(240,354)(241,355)(242,356)(243,357)(244,358)
(245,359)(246,360)(247,361)(248,362)(249,363)(250,364)(251,365)(252,366)
(253,367)(254,368)(255,369)(256,370)(257,371)(258,372)(259,373)(260,374)
(261,375)(262,376)(263,377)(264,378)(265,379)(266,380)(267,381)(268,382)
(269,383)(270,384)(271,385)(272,386)(273,387)(274,388)(275,389)(276,390)
(277,391)(278,392)(279,393)(280,394)(281,395)(282,396)(283,397)(284,398)
(285,399)(286,400)(287,401)(288,402)(289,403)(290,404)(291,405)(292,406)
(293,407)(294,408)(295,409)(296,410)(297,411)(298,412)(299,413)(300,414)
(301,415)(302,416)(303,417)(304,418)(305,419)(306,420)(307,421)(308,422)
(309,423)(310,424)(311,425)(312,426)(313,427)(314,428)(315,429)(316,430)
(317,431)(318,432)(319,433)(320,434)(321,435)(322,436)(323,437)(324,438)
(325,439)(326,440)(327,441)(328,442)(329,443)(330,444)(331,445)(332,446)
(333,447)(334,448)(335,449)(336,450)(337,451)(338,452)(339,453)(340,454)
(341,455)(342,456);;
s1 := ( 1,229)( 2,247)( 3,246)( 4,245)( 5,244)( 6,243)( 7,242)( 8,241)
( 9,240)( 10,239)( 11,238)( 12,237)( 13,236)( 14,235)( 15,234)( 16,233)
( 17,232)( 18,231)( 19,230)( 20,267)( 21,285)( 22,284)( 23,283)( 24,282)
( 25,281)( 26,280)( 27,279)( 28,278)( 29,277)( 30,276)( 31,275)( 32,274)
( 33,273)( 34,272)( 35,271)( 36,270)( 37,269)( 38,268)( 39,248)( 40,266)
( 41,265)( 42,264)( 43,263)( 44,262)( 45,261)( 46,260)( 47,259)( 48,258)
( 49,257)( 50,256)( 51,255)( 52,254)( 53,253)( 54,252)( 55,251)( 56,250)
( 57,249)( 58,286)( 59,304)( 60,303)( 61,302)( 62,301)( 63,300)( 64,299)
( 65,298)( 66,297)( 67,296)( 68,295)( 69,294)( 70,293)( 71,292)( 72,291)
( 73,290)( 74,289)( 75,288)( 76,287)( 77,324)( 78,342)( 79,341)( 80,340)
( 81,339)( 82,338)( 83,337)( 84,336)( 85,335)( 86,334)( 87,333)( 88,332)
( 89,331)( 90,330)( 91,329)( 92,328)( 93,327)( 94,326)( 95,325)( 96,305)
( 97,323)( 98,322)( 99,321)(100,320)(101,319)(102,318)(103,317)(104,316)
(105,315)(106,314)(107,313)(108,312)(109,311)(110,310)(111,309)(112,308)
(113,307)(114,306)(115,343)(116,361)(117,360)(118,359)(119,358)(120,357)
(121,356)(122,355)(123,354)(124,353)(125,352)(126,351)(127,350)(128,349)
(129,348)(130,347)(131,346)(132,345)(133,344)(134,381)(135,399)(136,398)
(137,397)(138,396)(139,395)(140,394)(141,393)(142,392)(143,391)(144,390)
(145,389)(146,388)(147,387)(148,386)(149,385)(150,384)(151,383)(152,382)
(153,362)(154,380)(155,379)(156,378)(157,377)(158,376)(159,375)(160,374)
(161,373)(162,372)(163,371)(164,370)(165,369)(166,368)(167,367)(168,366)
(169,365)(170,364)(171,363)(172,400)(173,418)(174,417)(175,416)(176,415)
(177,414)(178,413)(179,412)(180,411)(181,410)(182,409)(183,408)(184,407)
(185,406)(186,405)(187,404)(188,403)(189,402)(190,401)(191,438)(192,456)
(193,455)(194,454)(195,453)(196,452)(197,451)(198,450)(199,449)(200,448)
(201,447)(202,446)(203,445)(204,444)(205,443)(206,442)(207,441)(208,440)
(209,439)(210,419)(211,437)(212,436)(213,435)(214,434)(215,433)(216,432)
(217,431)(218,430)(219,429)(220,428)(221,427)(222,426)(223,425)(224,424)
(225,423)(226,422)(227,421)(228,420);;
s2 := ( 1, 21)( 2, 20)( 3, 38)( 4, 37)( 5, 36)( 6, 35)( 7, 34)( 8, 33)
( 9, 32)( 10, 31)( 11, 30)( 12, 29)( 13, 28)( 14, 27)( 15, 26)( 16, 25)
( 17, 24)( 18, 23)( 19, 22)( 39, 40)( 41, 57)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58, 78)( 59, 77)( 60, 95)( 61, 94)
( 62, 93)( 63, 92)( 64, 91)( 65, 90)( 66, 89)( 67, 88)( 68, 87)( 69, 86)
( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)( 96, 97)
( 98,114)( 99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)
(115,135)(116,134)(117,152)(118,151)(119,150)(120,149)(121,148)(122,147)
(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139)
(131,138)(132,137)(133,136)(153,154)(155,171)(156,170)(157,169)(158,168)
(159,167)(160,166)(161,165)(162,164)(172,192)(173,191)(174,209)(175,208)
(176,207)(177,206)(178,205)(179,204)(180,203)(181,202)(182,201)(183,200)
(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)(210,211)
(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)(219,221)
(229,306)(230,305)(231,323)(232,322)(233,321)(234,320)(235,319)(236,318)
(237,317)(238,316)(239,315)(240,314)(241,313)(242,312)(243,311)(244,310)
(245,309)(246,308)(247,307)(248,287)(249,286)(250,304)(251,303)(252,302)
(253,301)(254,300)(255,299)(256,298)(257,297)(258,296)(259,295)(260,294)
(261,293)(262,292)(263,291)(264,290)(265,289)(266,288)(267,325)(268,324)
(269,342)(270,341)(271,340)(272,339)(273,338)(274,337)(275,336)(276,335)
(277,334)(278,333)(279,332)(280,331)(281,330)(282,329)(283,328)(284,327)
(285,326)(343,420)(344,419)(345,437)(346,436)(347,435)(348,434)(349,433)
(350,432)(351,431)(352,430)(353,429)(354,428)(355,427)(356,426)(357,425)
(358,424)(359,423)(360,422)(361,421)(362,401)(363,400)(364,418)(365,417)
(366,416)(367,415)(368,414)(369,413)(370,412)(371,411)(372,410)(373,409)
(374,408)(375,407)(376,406)(377,405)(378,404)(379,403)(380,402)(381,439)
(382,438)(383,456)(384,455)(385,454)(386,453)(387,452)(388,451)(389,450)
(390,449)(391,448)(392,447)(393,446)(394,445)(395,444)(396,443)(397,442)
(398,441)(399,440);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(456)!(229,343)(230,344)(231,345)(232,346)(233,347)(234,348)(235,349)
(236,350)(237,351)(238,352)(239,353)(240,354)(241,355)(242,356)(243,357)
(244,358)(245,359)(246,360)(247,361)(248,362)(249,363)(250,364)(251,365)
(252,366)(253,367)(254,368)(255,369)(256,370)(257,371)(258,372)(259,373)
(260,374)(261,375)(262,376)(263,377)(264,378)(265,379)(266,380)(267,381)
(268,382)(269,383)(270,384)(271,385)(272,386)(273,387)(274,388)(275,389)
(276,390)(277,391)(278,392)(279,393)(280,394)(281,395)(282,396)(283,397)
(284,398)(285,399)(286,400)(287,401)(288,402)(289,403)(290,404)(291,405)
(292,406)(293,407)(294,408)(295,409)(296,410)(297,411)(298,412)(299,413)
(300,414)(301,415)(302,416)(303,417)(304,418)(305,419)(306,420)(307,421)
(308,422)(309,423)(310,424)(311,425)(312,426)(313,427)(314,428)(315,429)
(316,430)(317,431)(318,432)(319,433)(320,434)(321,435)(322,436)(323,437)
(324,438)(325,439)(326,440)(327,441)(328,442)(329,443)(330,444)(331,445)
(332,446)(333,447)(334,448)(335,449)(336,450)(337,451)(338,452)(339,453)
(340,454)(341,455)(342,456);
s1 := Sym(456)!( 1,229)( 2,247)( 3,246)( 4,245)( 5,244)( 6,243)( 7,242)
( 8,241)( 9,240)( 10,239)( 11,238)( 12,237)( 13,236)( 14,235)( 15,234)
( 16,233)( 17,232)( 18,231)( 19,230)( 20,267)( 21,285)( 22,284)( 23,283)
( 24,282)( 25,281)( 26,280)( 27,279)( 28,278)( 29,277)( 30,276)( 31,275)
( 32,274)( 33,273)( 34,272)( 35,271)( 36,270)( 37,269)( 38,268)( 39,248)
( 40,266)( 41,265)( 42,264)( 43,263)( 44,262)( 45,261)( 46,260)( 47,259)
( 48,258)( 49,257)( 50,256)( 51,255)( 52,254)( 53,253)( 54,252)( 55,251)
( 56,250)( 57,249)( 58,286)( 59,304)( 60,303)( 61,302)( 62,301)( 63,300)
( 64,299)( 65,298)( 66,297)( 67,296)( 68,295)( 69,294)( 70,293)( 71,292)
( 72,291)( 73,290)( 74,289)( 75,288)( 76,287)( 77,324)( 78,342)( 79,341)
( 80,340)( 81,339)( 82,338)( 83,337)( 84,336)( 85,335)( 86,334)( 87,333)
( 88,332)( 89,331)( 90,330)( 91,329)( 92,328)( 93,327)( 94,326)( 95,325)
( 96,305)( 97,323)( 98,322)( 99,321)(100,320)(101,319)(102,318)(103,317)
(104,316)(105,315)(106,314)(107,313)(108,312)(109,311)(110,310)(111,309)
(112,308)(113,307)(114,306)(115,343)(116,361)(117,360)(118,359)(119,358)
(120,357)(121,356)(122,355)(123,354)(124,353)(125,352)(126,351)(127,350)
(128,349)(129,348)(130,347)(131,346)(132,345)(133,344)(134,381)(135,399)
(136,398)(137,397)(138,396)(139,395)(140,394)(141,393)(142,392)(143,391)
(144,390)(145,389)(146,388)(147,387)(148,386)(149,385)(150,384)(151,383)
(152,382)(153,362)(154,380)(155,379)(156,378)(157,377)(158,376)(159,375)
(160,374)(161,373)(162,372)(163,371)(164,370)(165,369)(166,368)(167,367)
(168,366)(169,365)(170,364)(171,363)(172,400)(173,418)(174,417)(175,416)
(176,415)(177,414)(178,413)(179,412)(180,411)(181,410)(182,409)(183,408)
(184,407)(185,406)(186,405)(187,404)(188,403)(189,402)(190,401)(191,438)
(192,456)(193,455)(194,454)(195,453)(196,452)(197,451)(198,450)(199,449)
(200,448)(201,447)(202,446)(203,445)(204,444)(205,443)(206,442)(207,441)
(208,440)(209,439)(210,419)(211,437)(212,436)(213,435)(214,434)(215,433)
(216,432)(217,431)(218,430)(219,429)(220,428)(221,427)(222,426)(223,425)
(224,424)(225,423)(226,422)(227,421)(228,420);
s2 := Sym(456)!( 1, 21)( 2, 20)( 3, 38)( 4, 37)( 5, 36)( 6, 35)( 7, 34)
( 8, 33)( 9, 32)( 10, 31)( 11, 30)( 12, 29)( 13, 28)( 14, 27)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 39, 40)( 41, 57)( 42, 56)( 43, 55)
( 44, 54)( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58, 78)( 59, 77)( 60, 95)
( 61, 94)( 62, 93)( 63, 92)( 64, 91)( 65, 90)( 66, 89)( 67, 88)( 68, 87)
( 69, 86)( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)
( 96, 97)( 98,114)( 99,113)(100,112)(101,111)(102,110)(103,109)(104,108)
(105,107)(115,135)(116,134)(117,152)(118,151)(119,150)(120,149)(121,148)
(122,147)(123,146)(124,145)(125,144)(126,143)(127,142)(128,141)(129,140)
(130,139)(131,138)(132,137)(133,136)(153,154)(155,171)(156,170)(157,169)
(158,168)(159,167)(160,166)(161,165)(162,164)(172,192)(173,191)(174,209)
(175,208)(176,207)(177,206)(178,205)(179,204)(180,203)(181,202)(182,201)
(183,200)(184,199)(185,198)(186,197)(187,196)(188,195)(189,194)(190,193)
(210,211)(212,228)(213,227)(214,226)(215,225)(216,224)(217,223)(218,222)
(219,221)(229,306)(230,305)(231,323)(232,322)(233,321)(234,320)(235,319)
(236,318)(237,317)(238,316)(239,315)(240,314)(241,313)(242,312)(243,311)
(244,310)(245,309)(246,308)(247,307)(248,287)(249,286)(250,304)(251,303)
(252,302)(253,301)(254,300)(255,299)(256,298)(257,297)(258,296)(259,295)
(260,294)(261,293)(262,292)(263,291)(264,290)(265,289)(266,288)(267,325)
(268,324)(269,342)(270,341)(271,340)(272,339)(273,338)(274,337)(275,336)
(276,335)(277,334)(278,333)(279,332)(280,331)(281,330)(282,329)(283,328)
(284,327)(285,326)(343,420)(344,419)(345,437)(346,436)(347,435)(348,434)
(349,433)(350,432)(351,431)(352,430)(353,429)(354,428)(355,427)(356,426)
(357,425)(358,424)(359,423)(360,422)(361,421)(362,401)(363,400)(364,418)
(365,417)(366,416)(367,415)(368,414)(369,413)(370,412)(371,411)(372,410)
(373,409)(374,408)(375,407)(376,406)(377,405)(378,404)(379,403)(380,402)
(381,439)(382,438)(383,456)(384,455)(385,454)(386,453)(387,452)(388,451)
(389,450)(390,449)(391,448)(392,447)(393,446)(394,445)(395,444)(396,443)
(397,442)(398,441)(399,440);
poly := sub<Sym(456)|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 >;
References : None.
to this polytope