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Polytope of Type {4,57}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,57}*912
if this polytope has a name.
Group : SmallGroup(912,209)
Rank : 3
Schlafli Type : {4,57}
Number of vertices, edges, etc : 8, 228, 114
Order of s0s1s2 : 114
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,57,2} of size 1824
Vertex Figure Of :
   {2,4,57} of size 1824
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,57}*456
   4-fold quotients : {2,57}*228
   12-fold quotients : {2,19}*76
   19-fold quotients : {4,3}*48
   38-fold quotients : {4,3}*24
   76-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,57}*1824, {4,114}*1824
Permutation Representation (GAP) :
s0 := (  1,231)(  2,232)(  3,229)(  4,230)(  5,235)(  6,236)(  7,233)(  8,234)
(  9,239)( 10,240)( 11,237)( 12,238)( 13,243)( 14,244)( 15,241)( 16,242)
( 17,247)( 18,248)( 19,245)( 20,246)( 21,251)( 22,252)( 23,249)( 24,250)
( 25,255)( 26,256)( 27,253)( 28,254)( 29,259)( 30,260)( 31,257)( 32,258)
( 33,263)( 34,264)( 35,261)( 36,262)( 37,267)( 38,268)( 39,265)( 40,266)
( 41,271)( 42,272)( 43,269)( 44,270)( 45,275)( 46,276)( 47,273)( 48,274)
( 49,279)( 50,280)( 51,277)( 52,278)( 53,283)( 54,284)( 55,281)( 56,282)
( 57,287)( 58,288)( 59,285)( 60,286)( 61,291)( 62,292)( 63,289)( 64,290)
( 65,295)( 66,296)( 67,293)( 68,294)( 69,299)( 70,300)( 71,297)( 72,298)
( 73,303)( 74,304)( 75,301)( 76,302)( 77,307)( 78,308)( 79,305)( 80,306)
( 81,311)( 82,312)( 83,309)( 84,310)( 85,315)( 86,316)( 87,313)( 88,314)
( 89,319)( 90,320)( 91,317)( 92,318)( 93,323)( 94,324)( 95,321)( 96,322)
( 97,327)( 98,328)( 99,325)(100,326)(101,331)(102,332)(103,329)(104,330)
(105,335)(106,336)(107,333)(108,334)(109,339)(110,340)(111,337)(112,338)
(113,343)(114,344)(115,341)(116,342)(117,347)(118,348)(119,345)(120,346)
(121,351)(122,352)(123,349)(124,350)(125,355)(126,356)(127,353)(128,354)
(129,359)(130,360)(131,357)(132,358)(133,363)(134,364)(135,361)(136,362)
(137,367)(138,368)(139,365)(140,366)(141,371)(142,372)(143,369)(144,370)
(145,375)(146,376)(147,373)(148,374)(149,379)(150,380)(151,377)(152,378)
(153,383)(154,384)(155,381)(156,382)(157,387)(158,388)(159,385)(160,386)
(161,391)(162,392)(163,389)(164,390)(165,395)(166,396)(167,393)(168,394)
(169,399)(170,400)(171,397)(172,398)(173,403)(174,404)(175,401)(176,402)
(177,407)(178,408)(179,405)(180,406)(181,411)(182,412)(183,409)(184,410)
(185,415)(186,416)(187,413)(188,414)(189,419)(190,420)(191,417)(192,418)
(193,423)(194,424)(195,421)(196,422)(197,427)(198,428)(199,425)(200,426)
(201,431)(202,432)(203,429)(204,430)(205,435)(206,436)(207,433)(208,434)
(209,439)(210,440)(211,437)(212,438)(213,443)(214,444)(215,441)(216,442)
(217,447)(218,448)(219,445)(220,446)(221,451)(222,452)(223,449)(224,450)
(225,455)(226,456)(227,453)(228,454);;
s1 := (  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);;
s2 := (  1,157)(  2,160)(  3,159)(  4,158)(  5,153)(  6,156)(  7,155)(  8,154)
(  9,225)( 10,228)( 11,227)( 12,226)( 13,221)( 14,224)( 15,223)( 16,222)
( 17,217)( 18,220)( 19,219)( 20,218)( 21,213)( 22,216)( 23,215)( 24,214)
( 25,209)( 26,212)( 27,211)( 28,210)( 29,205)( 30,208)( 31,207)( 32,206)
( 33,201)( 34,204)( 35,203)( 36,202)( 37,197)( 38,200)( 39,199)( 40,198)
( 41,193)( 42,196)( 43,195)( 44,194)( 45,189)( 46,192)( 47,191)( 48,190)
( 49,185)( 50,188)( 51,187)( 52,186)( 53,181)( 54,184)( 55,183)( 56,182)
( 57,177)( 58,180)( 59,179)( 60,178)( 61,173)( 62,176)( 63,175)( 64,174)
( 65,169)( 66,172)( 67,171)( 68,170)( 69,165)( 70,168)( 71,167)( 72,166)
( 73,161)( 74,164)( 75,163)( 76,162)( 77, 81)( 78, 84)( 79, 83)( 80, 82)
( 85,149)( 86,152)( 87,151)( 88,150)( 89,145)( 90,148)( 91,147)( 92,146)
( 93,141)( 94,144)( 95,143)( 96,142)( 97,137)( 98,140)( 99,139)(100,138)
(101,133)(102,136)(103,135)(104,134)(105,129)(106,132)(107,131)(108,130)
(109,125)(110,128)(111,127)(112,126)(113,121)(114,124)(115,123)(116,122)
(118,120)(229,385)(230,388)(231,387)(232,386)(233,381)(234,384)(235,383)
(236,382)(237,453)(238,456)(239,455)(240,454)(241,449)(242,452)(243,451)
(244,450)(245,445)(246,448)(247,447)(248,446)(249,441)(250,444)(251,443)
(252,442)(253,437)(254,440)(255,439)(256,438)(257,433)(258,436)(259,435)
(260,434)(261,429)(262,432)(263,431)(264,430)(265,425)(266,428)(267,427)
(268,426)(269,421)(270,424)(271,423)(272,422)(273,417)(274,420)(275,419)
(276,418)(277,413)(278,416)(279,415)(280,414)(281,409)(282,412)(283,411)
(284,410)(285,405)(286,408)(287,407)(288,406)(289,401)(290,404)(291,403)
(292,402)(293,397)(294,400)(295,399)(296,398)(297,393)(298,396)(299,395)
(300,394)(301,389)(302,392)(303,391)(304,390)(305,309)(306,312)(307,311)
(308,310)(313,377)(314,380)(315,379)(316,378)(317,373)(318,376)(319,375)
(320,374)(321,369)(322,372)(323,371)(324,370)(325,365)(326,368)(327,367)
(328,366)(329,361)(330,364)(331,363)(332,362)(333,357)(334,360)(335,359)
(336,358)(337,353)(338,356)(339,355)(340,354)(341,349)(342,352)(343,351)
(344,350)(346,348);;
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*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(456)!(  1,231)(  2,232)(  3,229)(  4,230)(  5,235)(  6,236)(  7,233)
(  8,234)(  9,239)( 10,240)( 11,237)( 12,238)( 13,243)( 14,244)( 15,241)
( 16,242)( 17,247)( 18,248)( 19,245)( 20,246)( 21,251)( 22,252)( 23,249)
( 24,250)( 25,255)( 26,256)( 27,253)( 28,254)( 29,259)( 30,260)( 31,257)
( 32,258)( 33,263)( 34,264)( 35,261)( 36,262)( 37,267)( 38,268)( 39,265)
( 40,266)( 41,271)( 42,272)( 43,269)( 44,270)( 45,275)( 46,276)( 47,273)
( 48,274)( 49,279)( 50,280)( 51,277)( 52,278)( 53,283)( 54,284)( 55,281)
( 56,282)( 57,287)( 58,288)( 59,285)( 60,286)( 61,291)( 62,292)( 63,289)
( 64,290)( 65,295)( 66,296)( 67,293)( 68,294)( 69,299)( 70,300)( 71,297)
( 72,298)( 73,303)( 74,304)( 75,301)( 76,302)( 77,307)( 78,308)( 79,305)
( 80,306)( 81,311)( 82,312)( 83,309)( 84,310)( 85,315)( 86,316)( 87,313)
( 88,314)( 89,319)( 90,320)( 91,317)( 92,318)( 93,323)( 94,324)( 95,321)
( 96,322)( 97,327)( 98,328)( 99,325)(100,326)(101,331)(102,332)(103,329)
(104,330)(105,335)(106,336)(107,333)(108,334)(109,339)(110,340)(111,337)
(112,338)(113,343)(114,344)(115,341)(116,342)(117,347)(118,348)(119,345)
(120,346)(121,351)(122,352)(123,349)(124,350)(125,355)(126,356)(127,353)
(128,354)(129,359)(130,360)(131,357)(132,358)(133,363)(134,364)(135,361)
(136,362)(137,367)(138,368)(139,365)(140,366)(141,371)(142,372)(143,369)
(144,370)(145,375)(146,376)(147,373)(148,374)(149,379)(150,380)(151,377)
(152,378)(153,383)(154,384)(155,381)(156,382)(157,387)(158,388)(159,385)
(160,386)(161,391)(162,392)(163,389)(164,390)(165,395)(166,396)(167,393)
(168,394)(169,399)(170,400)(171,397)(172,398)(173,403)(174,404)(175,401)
(176,402)(177,407)(178,408)(179,405)(180,406)(181,411)(182,412)(183,409)
(184,410)(185,415)(186,416)(187,413)(188,414)(189,419)(190,420)(191,417)
(192,418)(193,423)(194,424)(195,421)(196,422)(197,427)(198,428)(199,425)
(200,426)(201,431)(202,432)(203,429)(204,430)(205,435)(206,436)(207,433)
(208,434)(209,439)(210,440)(211,437)(212,438)(213,443)(214,444)(215,441)
(216,442)(217,447)(218,448)(219,445)(220,446)(221,451)(222,452)(223,449)
(224,450)(225,455)(226,456)(227,453)(228,454);
s1 := 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);
s2 := Sym(456)!(  1,157)(  2,160)(  3,159)(  4,158)(  5,153)(  6,156)(  7,155)
(  8,154)(  9,225)( 10,228)( 11,227)( 12,226)( 13,221)( 14,224)( 15,223)
( 16,222)( 17,217)( 18,220)( 19,219)( 20,218)( 21,213)( 22,216)( 23,215)
( 24,214)( 25,209)( 26,212)( 27,211)( 28,210)( 29,205)( 30,208)( 31,207)
( 32,206)( 33,201)( 34,204)( 35,203)( 36,202)( 37,197)( 38,200)( 39,199)
( 40,198)( 41,193)( 42,196)( 43,195)( 44,194)( 45,189)( 46,192)( 47,191)
( 48,190)( 49,185)( 50,188)( 51,187)( 52,186)( 53,181)( 54,184)( 55,183)
( 56,182)( 57,177)( 58,180)( 59,179)( 60,178)( 61,173)( 62,176)( 63,175)
( 64,174)( 65,169)( 66,172)( 67,171)( 68,170)( 69,165)( 70,168)( 71,167)
( 72,166)( 73,161)( 74,164)( 75,163)( 76,162)( 77, 81)( 78, 84)( 79, 83)
( 80, 82)( 85,149)( 86,152)( 87,151)( 88,150)( 89,145)( 90,148)( 91,147)
( 92,146)( 93,141)( 94,144)( 95,143)( 96,142)( 97,137)( 98,140)( 99,139)
(100,138)(101,133)(102,136)(103,135)(104,134)(105,129)(106,132)(107,131)
(108,130)(109,125)(110,128)(111,127)(112,126)(113,121)(114,124)(115,123)
(116,122)(118,120)(229,385)(230,388)(231,387)(232,386)(233,381)(234,384)
(235,383)(236,382)(237,453)(238,456)(239,455)(240,454)(241,449)(242,452)
(243,451)(244,450)(245,445)(246,448)(247,447)(248,446)(249,441)(250,444)
(251,443)(252,442)(253,437)(254,440)(255,439)(256,438)(257,433)(258,436)
(259,435)(260,434)(261,429)(262,432)(263,431)(264,430)(265,425)(266,428)
(267,427)(268,426)(269,421)(270,424)(271,423)(272,422)(273,417)(274,420)
(275,419)(276,418)(277,413)(278,416)(279,415)(280,414)(281,409)(282,412)
(283,411)(284,410)(285,405)(286,408)(287,407)(288,406)(289,401)(290,404)
(291,403)(292,402)(293,397)(294,400)(295,399)(296,398)(297,393)(298,396)
(299,395)(300,394)(301,389)(302,392)(303,391)(304,390)(305,309)(306,312)
(307,311)(308,310)(313,377)(314,380)(315,379)(316,378)(317,373)(318,376)
(319,375)(320,374)(321,369)(322,372)(323,371)(324,370)(325,365)(326,368)
(327,367)(328,366)(329,361)(330,364)(331,363)(332,362)(333,357)(334,360)
(335,359)(336,358)(337,353)(338,356)(339,355)(340,354)(341,349)(342,352)
(343,351)(344,350)(346,348);
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*s2*s1*s0*s1*s2*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 >; 
 
References : None.
to this polytope