Polytope of Type {57,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {57,4}*912
if this polytope has a name.
Group : SmallGroup(912,209)
Rank : 3
Schlafli Type : {57,4}
Number of vertices, edges, etc : 114, 228, 8
Order of s0s1s2 : 114
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{57,4,2} of size 1824
Vertex Figure Of :
{2,57,4} of size 1824
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {57,4}*456
4-fold quotients : {57,2}*228
12-fold quotients : {19,2}*76
19-fold quotients : {3,4}*48
38-fold quotients : {3,4}*24
76-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {57,8}*1824, {114,4}*1824
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 73)( 6, 75)( 7, 74)( 8, 76)( 9, 69)( 10, 71)( 11, 70)
( 12, 72)( 13, 65)( 14, 67)( 15, 66)( 16, 68)( 17, 61)( 18, 63)( 19, 62)
( 20, 64)( 21, 57)( 22, 59)( 23, 58)( 24, 60)( 25, 53)( 26, 55)( 27, 54)
( 28, 56)( 29, 49)( 30, 51)( 31, 50)( 32, 52)( 33, 45)( 34, 47)( 35, 46)
( 36, 48)( 37, 41)( 38, 43)( 39, 42)( 40, 44)( 77,153)( 78,155)( 79,154)
( 80,156)( 81,225)( 82,227)( 83,226)( 84,228)( 85,221)( 86,223)( 87,222)
( 88,224)( 89,217)( 90,219)( 91,218)( 92,220)( 93,213)( 94,215)( 95,214)
( 96,216)( 97,209)( 98,211)( 99,210)(100,212)(101,205)(102,207)(103,206)
(104,208)(105,201)(106,203)(107,202)(108,204)(109,197)(110,199)(111,198)
(112,200)(113,193)(114,195)(115,194)(116,196)(117,189)(118,191)(119,190)
(120,192)(121,185)(122,187)(123,186)(124,188)(125,181)(126,183)(127,182)
(128,184)(129,177)(130,179)(131,178)(132,180)(133,173)(134,175)(135,174)
(136,176)(137,169)(138,171)(139,170)(140,172)(141,165)(142,167)(143,166)
(144,168)(145,161)(146,163)(147,162)(148,164)(149,157)(150,159)(151,158)
(152,160)(230,231)(233,301)(234,303)(235,302)(236,304)(237,297)(238,299)
(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)(246,291)
(247,290)(248,292)(249,285)(250,287)(251,286)(252,288)(253,281)(254,283)
(255,282)(256,284)(257,277)(258,279)(259,278)(260,280)(261,273)(262,275)
(263,274)(264,276)(265,269)(266,271)(267,270)(268,272)(305,381)(306,383)
(307,382)(308,384)(309,453)(310,455)(311,454)(312,456)(313,449)(314,451)
(315,450)(316,452)(317,445)(318,447)(319,446)(320,448)(321,441)(322,443)
(323,442)(324,444)(325,437)(326,439)(327,438)(328,440)(329,433)(330,435)
(331,434)(332,436)(333,429)(334,431)(335,430)(336,432)(337,425)(338,427)
(339,426)(340,428)(341,421)(342,423)(343,422)(344,424)(345,417)(346,419)
(347,418)(348,420)(349,413)(350,415)(351,414)(352,416)(353,409)(354,411)
(355,410)(356,412)(357,405)(358,407)(359,406)(360,408)(361,401)(362,403)
(363,402)(364,404)(365,397)(366,399)(367,398)(368,400)(369,393)(370,395)
(371,394)(372,396)(373,389)(374,391)(375,390)(376,392)(377,385)(378,387)
(379,386)(380,388);;
s1 := ( 1, 81)( 2, 82)( 3, 84)( 4, 83)( 5, 77)( 6, 78)( 7, 80)( 8, 79)
( 9,149)( 10,150)( 11,152)( 12,151)( 13,145)( 14,146)( 15,148)( 16,147)
( 17,141)( 18,142)( 19,144)( 20,143)( 21,137)( 22,138)( 23,140)( 24,139)
( 25,133)( 26,134)( 27,136)( 28,135)( 29,129)( 30,130)( 31,132)( 32,131)
( 33,125)( 34,126)( 35,128)( 36,127)( 37,121)( 38,122)( 39,124)( 40,123)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)( 48,115)
( 49,109)( 50,110)( 51,112)( 52,111)( 53,105)( 54,106)( 55,108)( 56,107)
( 57,101)( 58,102)( 59,104)( 60,103)( 61, 97)( 62, 98)( 63,100)( 64, 99)
( 65, 93)( 66, 94)( 67, 96)( 68, 95)( 69, 89)( 70, 90)( 71, 92)( 72, 91)
( 73, 85)( 74, 86)( 75, 88)( 76, 87)(153,157)(154,158)(155,160)(156,159)
(161,225)(162,226)(163,228)(164,227)(165,221)(166,222)(167,224)(168,223)
(169,217)(170,218)(171,220)(172,219)(173,213)(174,214)(175,216)(176,215)
(177,209)(178,210)(179,212)(180,211)(181,205)(182,206)(183,208)(184,207)
(185,201)(186,202)(187,204)(188,203)(189,197)(190,198)(191,200)(192,199)
(195,196)(229,309)(230,310)(231,312)(232,311)(233,305)(234,306)(235,308)
(236,307)(237,377)(238,378)(239,380)(240,379)(241,373)(242,374)(243,376)
(244,375)(245,369)(246,370)(247,372)(248,371)(249,365)(250,366)(251,368)
(252,367)(253,361)(254,362)(255,364)(256,363)(257,357)(258,358)(259,360)
(260,359)(261,353)(262,354)(263,356)(264,355)(265,349)(266,350)(267,352)
(268,351)(269,345)(270,346)(271,348)(272,347)(273,341)(274,342)(275,344)
(276,343)(277,337)(278,338)(279,340)(280,339)(281,333)(282,334)(283,336)
(284,335)(285,329)(286,330)(287,332)(288,331)(289,325)(290,326)(291,328)
(292,327)(293,321)(294,322)(295,324)(296,323)(297,317)(298,318)(299,320)
(300,319)(301,313)(302,314)(303,316)(304,315)(381,385)(382,386)(383,388)
(384,387)(389,453)(390,454)(391,456)(392,455)(393,449)(394,450)(395,452)
(396,451)(397,445)(398,446)(399,448)(400,447)(401,441)(402,442)(403,444)
(404,443)(405,437)(406,438)(407,440)(408,439)(409,433)(410,434)(411,436)
(412,435)(413,429)(414,430)(415,432)(416,431)(417,425)(418,426)(419,428)
(420,427)(423,424);;
s2 := ( 1,232)( 2,231)( 3,230)( 4,229)( 5,236)( 6,235)( 7,234)( 8,233)
( 9,240)( 10,239)( 11,238)( 12,237)( 13,244)( 14,243)( 15,242)( 16,241)
( 17,248)( 18,247)( 19,246)( 20,245)( 21,252)( 22,251)( 23,250)( 24,249)
( 25,256)( 26,255)( 27,254)( 28,253)( 29,260)( 30,259)( 31,258)( 32,257)
( 33,264)( 34,263)( 35,262)( 36,261)( 37,268)( 38,267)( 39,266)( 40,265)
( 41,272)( 42,271)( 43,270)( 44,269)( 45,276)( 46,275)( 47,274)( 48,273)
( 49,280)( 50,279)( 51,278)( 52,277)( 53,284)( 54,283)( 55,282)( 56,281)
( 57,288)( 58,287)( 59,286)( 60,285)( 61,292)( 62,291)( 63,290)( 64,289)
( 65,296)( 66,295)( 67,294)( 68,293)( 69,300)( 70,299)( 71,298)( 72,297)
( 73,304)( 74,303)( 75,302)( 76,301)( 77,308)( 78,307)( 79,306)( 80,305)
( 81,312)( 82,311)( 83,310)( 84,309)( 85,316)( 86,315)( 87,314)( 88,313)
( 89,320)( 90,319)( 91,318)( 92,317)( 93,324)( 94,323)( 95,322)( 96,321)
( 97,328)( 98,327)( 99,326)(100,325)(101,332)(102,331)(103,330)(104,329)
(105,336)(106,335)(107,334)(108,333)(109,340)(110,339)(111,338)(112,337)
(113,344)(114,343)(115,342)(116,341)(117,348)(118,347)(119,346)(120,345)
(121,352)(122,351)(123,350)(124,349)(125,356)(126,355)(127,354)(128,353)
(129,360)(130,359)(131,358)(132,357)(133,364)(134,363)(135,362)(136,361)
(137,368)(138,367)(139,366)(140,365)(141,372)(142,371)(143,370)(144,369)
(145,376)(146,375)(147,374)(148,373)(149,380)(150,379)(151,378)(152,377)
(153,384)(154,383)(155,382)(156,381)(157,388)(158,387)(159,386)(160,385)
(161,392)(162,391)(163,390)(164,389)(165,396)(166,395)(167,394)(168,393)
(169,400)(170,399)(171,398)(172,397)(173,404)(174,403)(175,402)(176,401)
(177,408)(178,407)(179,406)(180,405)(181,412)(182,411)(183,410)(184,409)
(185,416)(186,415)(187,414)(188,413)(189,420)(190,419)(191,418)(192,417)
(193,424)(194,423)(195,422)(196,421)(197,428)(198,427)(199,426)(200,425)
(201,432)(202,431)(203,430)(204,429)(205,436)(206,435)(207,434)(208,433)
(209,440)(210,439)(211,438)(212,437)(213,444)(214,443)(215,442)(216,441)
(217,448)(218,447)(219,446)(220,445)(221,452)(222,451)(223,450)(224,449)
(225,456)(226,455)(227,454)(228,453);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(456)!( 2, 3)( 5, 73)( 6, 75)( 7, 74)( 8, 76)( 9, 69)( 10, 71)
( 11, 70)( 12, 72)( 13, 65)( 14, 67)( 15, 66)( 16, 68)( 17, 61)( 18, 63)
( 19, 62)( 20, 64)( 21, 57)( 22, 59)( 23, 58)( 24, 60)( 25, 53)( 26, 55)
( 27, 54)( 28, 56)( 29, 49)( 30, 51)( 31, 50)( 32, 52)( 33, 45)( 34, 47)
( 35, 46)( 36, 48)( 37, 41)( 38, 43)( 39, 42)( 40, 44)( 77,153)( 78,155)
( 79,154)( 80,156)( 81,225)( 82,227)( 83,226)( 84,228)( 85,221)( 86,223)
( 87,222)( 88,224)( 89,217)( 90,219)( 91,218)( 92,220)( 93,213)( 94,215)
( 95,214)( 96,216)( 97,209)( 98,211)( 99,210)(100,212)(101,205)(102,207)
(103,206)(104,208)(105,201)(106,203)(107,202)(108,204)(109,197)(110,199)
(111,198)(112,200)(113,193)(114,195)(115,194)(116,196)(117,189)(118,191)
(119,190)(120,192)(121,185)(122,187)(123,186)(124,188)(125,181)(126,183)
(127,182)(128,184)(129,177)(130,179)(131,178)(132,180)(133,173)(134,175)
(135,174)(136,176)(137,169)(138,171)(139,170)(140,172)(141,165)(142,167)
(143,166)(144,168)(145,161)(146,163)(147,162)(148,164)(149,157)(150,159)
(151,158)(152,160)(230,231)(233,301)(234,303)(235,302)(236,304)(237,297)
(238,299)(239,298)(240,300)(241,293)(242,295)(243,294)(244,296)(245,289)
(246,291)(247,290)(248,292)(249,285)(250,287)(251,286)(252,288)(253,281)
(254,283)(255,282)(256,284)(257,277)(258,279)(259,278)(260,280)(261,273)
(262,275)(263,274)(264,276)(265,269)(266,271)(267,270)(268,272)(305,381)
(306,383)(307,382)(308,384)(309,453)(310,455)(311,454)(312,456)(313,449)
(314,451)(315,450)(316,452)(317,445)(318,447)(319,446)(320,448)(321,441)
(322,443)(323,442)(324,444)(325,437)(326,439)(327,438)(328,440)(329,433)
(330,435)(331,434)(332,436)(333,429)(334,431)(335,430)(336,432)(337,425)
(338,427)(339,426)(340,428)(341,421)(342,423)(343,422)(344,424)(345,417)
(346,419)(347,418)(348,420)(349,413)(350,415)(351,414)(352,416)(353,409)
(354,411)(355,410)(356,412)(357,405)(358,407)(359,406)(360,408)(361,401)
(362,403)(363,402)(364,404)(365,397)(366,399)(367,398)(368,400)(369,393)
(370,395)(371,394)(372,396)(373,389)(374,391)(375,390)(376,392)(377,385)
(378,387)(379,386)(380,388);
s1 := Sym(456)!( 1, 81)( 2, 82)( 3, 84)( 4, 83)( 5, 77)( 6, 78)( 7, 80)
( 8, 79)( 9,149)( 10,150)( 11,152)( 12,151)( 13,145)( 14,146)( 15,148)
( 16,147)( 17,141)( 18,142)( 19,144)( 20,143)( 21,137)( 22,138)( 23,140)
( 24,139)( 25,133)( 26,134)( 27,136)( 28,135)( 29,129)( 30,130)( 31,132)
( 32,131)( 33,125)( 34,126)( 35,128)( 36,127)( 37,121)( 38,122)( 39,124)
( 40,123)( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)
( 48,115)( 49,109)( 50,110)( 51,112)( 52,111)( 53,105)( 54,106)( 55,108)
( 56,107)( 57,101)( 58,102)( 59,104)( 60,103)( 61, 97)( 62, 98)( 63,100)
( 64, 99)( 65, 93)( 66, 94)( 67, 96)( 68, 95)( 69, 89)( 70, 90)( 71, 92)
( 72, 91)( 73, 85)( 74, 86)( 75, 88)( 76, 87)(153,157)(154,158)(155,160)
(156,159)(161,225)(162,226)(163,228)(164,227)(165,221)(166,222)(167,224)
(168,223)(169,217)(170,218)(171,220)(172,219)(173,213)(174,214)(175,216)
(176,215)(177,209)(178,210)(179,212)(180,211)(181,205)(182,206)(183,208)
(184,207)(185,201)(186,202)(187,204)(188,203)(189,197)(190,198)(191,200)
(192,199)(195,196)(229,309)(230,310)(231,312)(232,311)(233,305)(234,306)
(235,308)(236,307)(237,377)(238,378)(239,380)(240,379)(241,373)(242,374)
(243,376)(244,375)(245,369)(246,370)(247,372)(248,371)(249,365)(250,366)
(251,368)(252,367)(253,361)(254,362)(255,364)(256,363)(257,357)(258,358)
(259,360)(260,359)(261,353)(262,354)(263,356)(264,355)(265,349)(266,350)
(267,352)(268,351)(269,345)(270,346)(271,348)(272,347)(273,341)(274,342)
(275,344)(276,343)(277,337)(278,338)(279,340)(280,339)(281,333)(282,334)
(283,336)(284,335)(285,329)(286,330)(287,332)(288,331)(289,325)(290,326)
(291,328)(292,327)(293,321)(294,322)(295,324)(296,323)(297,317)(298,318)
(299,320)(300,319)(301,313)(302,314)(303,316)(304,315)(381,385)(382,386)
(383,388)(384,387)(389,453)(390,454)(391,456)(392,455)(393,449)(394,450)
(395,452)(396,451)(397,445)(398,446)(399,448)(400,447)(401,441)(402,442)
(403,444)(404,443)(405,437)(406,438)(407,440)(408,439)(409,433)(410,434)
(411,436)(412,435)(413,429)(414,430)(415,432)(416,431)(417,425)(418,426)
(419,428)(420,427)(423,424);
s2 := Sym(456)!( 1,232)( 2,231)( 3,230)( 4,229)( 5,236)( 6,235)( 7,234)
( 8,233)( 9,240)( 10,239)( 11,238)( 12,237)( 13,244)( 14,243)( 15,242)
( 16,241)( 17,248)( 18,247)( 19,246)( 20,245)( 21,252)( 22,251)( 23,250)
( 24,249)( 25,256)( 26,255)( 27,254)( 28,253)( 29,260)( 30,259)( 31,258)
( 32,257)( 33,264)( 34,263)( 35,262)( 36,261)( 37,268)( 38,267)( 39,266)
( 40,265)( 41,272)( 42,271)( 43,270)( 44,269)( 45,276)( 46,275)( 47,274)
( 48,273)( 49,280)( 50,279)( 51,278)( 52,277)( 53,284)( 54,283)( 55,282)
( 56,281)( 57,288)( 58,287)( 59,286)( 60,285)( 61,292)( 62,291)( 63,290)
( 64,289)( 65,296)( 66,295)( 67,294)( 68,293)( 69,300)( 70,299)( 71,298)
( 72,297)( 73,304)( 74,303)( 75,302)( 76,301)( 77,308)( 78,307)( 79,306)
( 80,305)( 81,312)( 82,311)( 83,310)( 84,309)( 85,316)( 86,315)( 87,314)
( 88,313)( 89,320)( 90,319)( 91,318)( 92,317)( 93,324)( 94,323)( 95,322)
( 96,321)( 97,328)( 98,327)( 99,326)(100,325)(101,332)(102,331)(103,330)
(104,329)(105,336)(106,335)(107,334)(108,333)(109,340)(110,339)(111,338)
(112,337)(113,344)(114,343)(115,342)(116,341)(117,348)(118,347)(119,346)
(120,345)(121,352)(122,351)(123,350)(124,349)(125,356)(126,355)(127,354)
(128,353)(129,360)(130,359)(131,358)(132,357)(133,364)(134,363)(135,362)
(136,361)(137,368)(138,367)(139,366)(140,365)(141,372)(142,371)(143,370)
(144,369)(145,376)(146,375)(147,374)(148,373)(149,380)(150,379)(151,378)
(152,377)(153,384)(154,383)(155,382)(156,381)(157,388)(158,387)(159,386)
(160,385)(161,392)(162,391)(163,390)(164,389)(165,396)(166,395)(167,394)
(168,393)(169,400)(170,399)(171,398)(172,397)(173,404)(174,403)(175,402)
(176,401)(177,408)(178,407)(179,406)(180,405)(181,412)(182,411)(183,410)
(184,409)(185,416)(186,415)(187,414)(188,413)(189,420)(190,419)(191,418)
(192,417)(193,424)(194,423)(195,422)(196,421)(197,428)(198,427)(199,426)
(200,425)(201,432)(202,431)(203,430)(204,429)(205,436)(206,435)(207,434)
(208,433)(209,440)(210,439)(211,438)(212,437)(213,444)(214,443)(215,442)
(216,441)(217,448)(218,447)(219,446)(220,445)(221,452)(222,451)(223,450)
(224,449)(225,456)(226,455)(227,454)(228,453);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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