Polytope of Type {2,57,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,57,4}*1824
if this polytope has a name.
Group : SmallGroup(1824,1247)
Rank : 4
Schlafli Type : {2,57,4}
Number of vertices, edges, etc : 2, 114, 228, 8
Order of s0s1s2s3 : 114
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
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 : {2,57,4}*912
4-fold quotients : {2,57,2}*456
12-fold quotients : {2,19,2}*152
19-fold quotients : {2,3,4}*96
38-fold quotients : {2,3,4}*48
76-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 75)( 8, 77)( 9, 76)( 10, 78)( 11, 71)( 12, 73)( 13, 72)
( 14, 74)( 15, 67)( 16, 69)( 17, 68)( 18, 70)( 19, 63)( 20, 65)( 21, 64)
( 22, 66)( 23, 59)( 24, 61)( 25, 60)( 26, 62)( 27, 55)( 28, 57)( 29, 56)
( 30, 58)( 31, 51)( 32, 53)( 33, 52)( 34, 54)( 35, 47)( 36, 49)( 37, 48)
( 38, 50)( 39, 43)( 40, 45)( 41, 44)( 42, 46)( 79,155)( 80,157)( 81,156)
( 82,158)( 83,227)( 84,229)( 85,228)( 86,230)( 87,223)( 88,225)( 89,224)
( 90,226)( 91,219)( 92,221)( 93,220)( 94,222)( 95,215)( 96,217)( 97,216)
( 98,218)( 99,211)(100,213)(101,212)(102,214)(103,207)(104,209)(105,208)
(106,210)(107,203)(108,205)(109,204)(110,206)(111,199)(112,201)(113,200)
(114,202)(115,195)(116,197)(117,196)(118,198)(119,191)(120,193)(121,192)
(122,194)(123,187)(124,189)(125,188)(126,190)(127,183)(128,185)(129,184)
(130,186)(131,179)(132,181)(133,180)(134,182)(135,175)(136,177)(137,176)
(138,178)(139,171)(140,173)(141,172)(142,174)(143,167)(144,169)(145,168)
(146,170)(147,163)(148,165)(149,164)(150,166)(151,159)(152,161)(153,160)
(154,162)(232,233)(235,303)(236,305)(237,304)(238,306)(239,299)(240,301)
(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)(248,293)
(249,292)(250,294)(251,287)(252,289)(253,288)(254,290)(255,283)(256,285)
(257,284)(258,286)(259,279)(260,281)(261,280)(262,282)(263,275)(264,277)
(265,276)(266,278)(267,271)(268,273)(269,272)(270,274)(307,383)(308,385)
(309,384)(310,386)(311,455)(312,457)(313,456)(314,458)(315,451)(316,453)
(317,452)(318,454)(319,447)(320,449)(321,448)(322,450)(323,443)(324,445)
(325,444)(326,446)(327,439)(328,441)(329,440)(330,442)(331,435)(332,437)
(333,436)(334,438)(335,431)(336,433)(337,432)(338,434)(339,427)(340,429)
(341,428)(342,430)(343,423)(344,425)(345,424)(346,426)(347,419)(348,421)
(349,420)(350,422)(351,415)(352,417)(353,416)(354,418)(355,411)(356,413)
(357,412)(358,414)(359,407)(360,409)(361,408)(362,410)(363,403)(364,405)
(365,404)(366,406)(367,399)(368,401)(369,400)(370,402)(371,395)(372,397)
(373,396)(374,398)(375,391)(376,393)(377,392)(378,394)(379,387)(380,389)
(381,388)(382,390);;
s2 := ( 3, 83)( 4, 84)( 5, 86)( 6, 85)( 7, 79)( 8, 80)( 9, 82)( 10, 81)
( 11,151)( 12,152)( 13,154)( 14,153)( 15,147)( 16,148)( 17,150)( 18,149)
( 19,143)( 20,144)( 21,146)( 22,145)( 23,139)( 24,140)( 25,142)( 26,141)
( 27,135)( 28,136)( 29,138)( 30,137)( 31,131)( 32,132)( 33,134)( 34,133)
( 35,127)( 36,128)( 37,130)( 38,129)( 39,123)( 40,124)( 41,126)( 42,125)
( 43,119)( 44,120)( 45,122)( 46,121)( 47,115)( 48,116)( 49,118)( 50,117)
( 51,111)( 52,112)( 53,114)( 54,113)( 55,107)( 56,108)( 57,110)( 58,109)
( 59,103)( 60,104)( 61,106)( 62,105)( 63, 99)( 64,100)( 65,102)( 66,101)
( 67, 95)( 68, 96)( 69, 98)( 70, 97)( 71, 91)( 72, 92)( 73, 94)( 74, 93)
( 75, 87)( 76, 88)( 77, 90)( 78, 89)(155,159)(156,160)(157,162)(158,161)
(163,227)(164,228)(165,230)(166,229)(167,223)(168,224)(169,226)(170,225)
(171,219)(172,220)(173,222)(174,221)(175,215)(176,216)(177,218)(178,217)
(179,211)(180,212)(181,214)(182,213)(183,207)(184,208)(185,210)(186,209)
(187,203)(188,204)(189,206)(190,205)(191,199)(192,200)(193,202)(194,201)
(197,198)(231,311)(232,312)(233,314)(234,313)(235,307)(236,308)(237,310)
(238,309)(239,379)(240,380)(241,382)(242,381)(243,375)(244,376)(245,378)
(246,377)(247,371)(248,372)(249,374)(250,373)(251,367)(252,368)(253,370)
(254,369)(255,363)(256,364)(257,366)(258,365)(259,359)(260,360)(261,362)
(262,361)(263,355)(264,356)(265,358)(266,357)(267,351)(268,352)(269,354)
(270,353)(271,347)(272,348)(273,350)(274,349)(275,343)(276,344)(277,346)
(278,345)(279,339)(280,340)(281,342)(282,341)(283,335)(284,336)(285,338)
(286,337)(287,331)(288,332)(289,334)(290,333)(291,327)(292,328)(293,330)
(294,329)(295,323)(296,324)(297,326)(298,325)(299,319)(300,320)(301,322)
(302,321)(303,315)(304,316)(305,318)(306,317)(383,387)(384,388)(385,390)
(386,389)(391,455)(392,456)(393,458)(394,457)(395,451)(396,452)(397,454)
(398,453)(399,447)(400,448)(401,450)(402,449)(403,443)(404,444)(405,446)
(406,445)(407,439)(408,440)(409,442)(410,441)(411,435)(412,436)(413,438)
(414,437)(415,431)(416,432)(417,434)(418,433)(419,427)(420,428)(421,430)
(422,429)(425,426);;
s3 := ( 3,234)( 4,233)( 5,232)( 6,231)( 7,238)( 8,237)( 9,236)( 10,235)
( 11,242)( 12,241)( 13,240)( 14,239)( 15,246)( 16,245)( 17,244)( 18,243)
( 19,250)( 20,249)( 21,248)( 22,247)( 23,254)( 24,253)( 25,252)( 26,251)
( 27,258)( 28,257)( 29,256)( 30,255)( 31,262)( 32,261)( 33,260)( 34,259)
( 35,266)( 36,265)( 37,264)( 38,263)( 39,270)( 40,269)( 41,268)( 42,267)
( 43,274)( 44,273)( 45,272)( 46,271)( 47,278)( 48,277)( 49,276)( 50,275)
( 51,282)( 52,281)( 53,280)( 54,279)( 55,286)( 56,285)( 57,284)( 58,283)
( 59,290)( 60,289)( 61,288)( 62,287)( 63,294)( 64,293)( 65,292)( 66,291)
( 67,298)( 68,297)( 69,296)( 70,295)( 71,302)( 72,301)( 73,300)( 74,299)
( 75,306)( 76,305)( 77,304)( 78,303)( 79,310)( 80,309)( 81,308)( 82,307)
( 83,314)( 84,313)( 85,312)( 86,311)( 87,318)( 88,317)( 89,316)( 90,315)
( 91,322)( 92,321)( 93,320)( 94,319)( 95,326)( 96,325)( 97,324)( 98,323)
( 99,330)(100,329)(101,328)(102,327)(103,334)(104,333)(105,332)(106,331)
(107,338)(108,337)(109,336)(110,335)(111,342)(112,341)(113,340)(114,339)
(115,346)(116,345)(117,344)(118,343)(119,350)(120,349)(121,348)(122,347)
(123,354)(124,353)(125,352)(126,351)(127,358)(128,357)(129,356)(130,355)
(131,362)(132,361)(133,360)(134,359)(135,366)(136,365)(137,364)(138,363)
(139,370)(140,369)(141,368)(142,367)(143,374)(144,373)(145,372)(146,371)
(147,378)(148,377)(149,376)(150,375)(151,382)(152,381)(153,380)(154,379)
(155,386)(156,385)(157,384)(158,383)(159,390)(160,389)(161,388)(162,387)
(163,394)(164,393)(165,392)(166,391)(167,398)(168,397)(169,396)(170,395)
(171,402)(172,401)(173,400)(174,399)(175,406)(176,405)(177,404)(178,403)
(179,410)(180,409)(181,408)(182,407)(183,414)(184,413)(185,412)(186,411)
(187,418)(188,417)(189,416)(190,415)(191,422)(192,421)(193,420)(194,419)
(195,426)(196,425)(197,424)(198,423)(199,430)(200,429)(201,428)(202,427)
(203,434)(204,433)(205,432)(206,431)(207,438)(208,437)(209,436)(210,435)
(211,442)(212,441)(213,440)(214,439)(215,446)(216,445)(217,444)(218,443)
(219,450)(220,449)(221,448)(222,447)(223,454)(224,453)(225,452)(226,451)
(227,458)(228,457)(229,456)(230,455);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(458)!(1,2);
s1 := Sym(458)!( 4, 5)( 7, 75)( 8, 77)( 9, 76)( 10, 78)( 11, 71)( 12, 73)
( 13, 72)( 14, 74)( 15, 67)( 16, 69)( 17, 68)( 18, 70)( 19, 63)( 20, 65)
( 21, 64)( 22, 66)( 23, 59)( 24, 61)( 25, 60)( 26, 62)( 27, 55)( 28, 57)
( 29, 56)( 30, 58)( 31, 51)( 32, 53)( 33, 52)( 34, 54)( 35, 47)( 36, 49)
( 37, 48)( 38, 50)( 39, 43)( 40, 45)( 41, 44)( 42, 46)( 79,155)( 80,157)
( 81,156)( 82,158)( 83,227)( 84,229)( 85,228)( 86,230)( 87,223)( 88,225)
( 89,224)( 90,226)( 91,219)( 92,221)( 93,220)( 94,222)( 95,215)( 96,217)
( 97,216)( 98,218)( 99,211)(100,213)(101,212)(102,214)(103,207)(104,209)
(105,208)(106,210)(107,203)(108,205)(109,204)(110,206)(111,199)(112,201)
(113,200)(114,202)(115,195)(116,197)(117,196)(118,198)(119,191)(120,193)
(121,192)(122,194)(123,187)(124,189)(125,188)(126,190)(127,183)(128,185)
(129,184)(130,186)(131,179)(132,181)(133,180)(134,182)(135,175)(136,177)
(137,176)(138,178)(139,171)(140,173)(141,172)(142,174)(143,167)(144,169)
(145,168)(146,170)(147,163)(148,165)(149,164)(150,166)(151,159)(152,161)
(153,160)(154,162)(232,233)(235,303)(236,305)(237,304)(238,306)(239,299)
(240,301)(241,300)(242,302)(243,295)(244,297)(245,296)(246,298)(247,291)
(248,293)(249,292)(250,294)(251,287)(252,289)(253,288)(254,290)(255,283)
(256,285)(257,284)(258,286)(259,279)(260,281)(261,280)(262,282)(263,275)
(264,277)(265,276)(266,278)(267,271)(268,273)(269,272)(270,274)(307,383)
(308,385)(309,384)(310,386)(311,455)(312,457)(313,456)(314,458)(315,451)
(316,453)(317,452)(318,454)(319,447)(320,449)(321,448)(322,450)(323,443)
(324,445)(325,444)(326,446)(327,439)(328,441)(329,440)(330,442)(331,435)
(332,437)(333,436)(334,438)(335,431)(336,433)(337,432)(338,434)(339,427)
(340,429)(341,428)(342,430)(343,423)(344,425)(345,424)(346,426)(347,419)
(348,421)(349,420)(350,422)(351,415)(352,417)(353,416)(354,418)(355,411)
(356,413)(357,412)(358,414)(359,407)(360,409)(361,408)(362,410)(363,403)
(364,405)(365,404)(366,406)(367,399)(368,401)(369,400)(370,402)(371,395)
(372,397)(373,396)(374,398)(375,391)(376,393)(377,392)(378,394)(379,387)
(380,389)(381,388)(382,390);
s2 := Sym(458)!( 3, 83)( 4, 84)( 5, 86)( 6, 85)( 7, 79)( 8, 80)( 9, 82)
( 10, 81)( 11,151)( 12,152)( 13,154)( 14,153)( 15,147)( 16,148)( 17,150)
( 18,149)( 19,143)( 20,144)( 21,146)( 22,145)( 23,139)( 24,140)( 25,142)
( 26,141)( 27,135)( 28,136)( 29,138)( 30,137)( 31,131)( 32,132)( 33,134)
( 34,133)( 35,127)( 36,128)( 37,130)( 38,129)( 39,123)( 40,124)( 41,126)
( 42,125)( 43,119)( 44,120)( 45,122)( 46,121)( 47,115)( 48,116)( 49,118)
( 50,117)( 51,111)( 52,112)( 53,114)( 54,113)( 55,107)( 56,108)( 57,110)
( 58,109)( 59,103)( 60,104)( 61,106)( 62,105)( 63, 99)( 64,100)( 65,102)
( 66,101)( 67, 95)( 68, 96)( 69, 98)( 70, 97)( 71, 91)( 72, 92)( 73, 94)
( 74, 93)( 75, 87)( 76, 88)( 77, 90)( 78, 89)(155,159)(156,160)(157,162)
(158,161)(163,227)(164,228)(165,230)(166,229)(167,223)(168,224)(169,226)
(170,225)(171,219)(172,220)(173,222)(174,221)(175,215)(176,216)(177,218)
(178,217)(179,211)(180,212)(181,214)(182,213)(183,207)(184,208)(185,210)
(186,209)(187,203)(188,204)(189,206)(190,205)(191,199)(192,200)(193,202)
(194,201)(197,198)(231,311)(232,312)(233,314)(234,313)(235,307)(236,308)
(237,310)(238,309)(239,379)(240,380)(241,382)(242,381)(243,375)(244,376)
(245,378)(246,377)(247,371)(248,372)(249,374)(250,373)(251,367)(252,368)
(253,370)(254,369)(255,363)(256,364)(257,366)(258,365)(259,359)(260,360)
(261,362)(262,361)(263,355)(264,356)(265,358)(266,357)(267,351)(268,352)
(269,354)(270,353)(271,347)(272,348)(273,350)(274,349)(275,343)(276,344)
(277,346)(278,345)(279,339)(280,340)(281,342)(282,341)(283,335)(284,336)
(285,338)(286,337)(287,331)(288,332)(289,334)(290,333)(291,327)(292,328)
(293,330)(294,329)(295,323)(296,324)(297,326)(298,325)(299,319)(300,320)
(301,322)(302,321)(303,315)(304,316)(305,318)(306,317)(383,387)(384,388)
(385,390)(386,389)(391,455)(392,456)(393,458)(394,457)(395,451)(396,452)
(397,454)(398,453)(399,447)(400,448)(401,450)(402,449)(403,443)(404,444)
(405,446)(406,445)(407,439)(408,440)(409,442)(410,441)(411,435)(412,436)
(413,438)(414,437)(415,431)(416,432)(417,434)(418,433)(419,427)(420,428)
(421,430)(422,429)(425,426);
s3 := Sym(458)!( 3,234)( 4,233)( 5,232)( 6,231)( 7,238)( 8,237)( 9,236)
( 10,235)( 11,242)( 12,241)( 13,240)( 14,239)( 15,246)( 16,245)( 17,244)
( 18,243)( 19,250)( 20,249)( 21,248)( 22,247)( 23,254)( 24,253)( 25,252)
( 26,251)( 27,258)( 28,257)( 29,256)( 30,255)( 31,262)( 32,261)( 33,260)
( 34,259)( 35,266)( 36,265)( 37,264)( 38,263)( 39,270)( 40,269)( 41,268)
( 42,267)( 43,274)( 44,273)( 45,272)( 46,271)( 47,278)( 48,277)( 49,276)
( 50,275)( 51,282)( 52,281)( 53,280)( 54,279)( 55,286)( 56,285)( 57,284)
( 58,283)( 59,290)( 60,289)( 61,288)( 62,287)( 63,294)( 64,293)( 65,292)
( 66,291)( 67,298)( 68,297)( 69,296)( 70,295)( 71,302)( 72,301)( 73,300)
( 74,299)( 75,306)( 76,305)( 77,304)( 78,303)( 79,310)( 80,309)( 81,308)
( 82,307)( 83,314)( 84,313)( 85,312)( 86,311)( 87,318)( 88,317)( 89,316)
( 90,315)( 91,322)( 92,321)( 93,320)( 94,319)( 95,326)( 96,325)( 97,324)
( 98,323)( 99,330)(100,329)(101,328)(102,327)(103,334)(104,333)(105,332)
(106,331)(107,338)(108,337)(109,336)(110,335)(111,342)(112,341)(113,340)
(114,339)(115,346)(116,345)(117,344)(118,343)(119,350)(120,349)(121,348)
(122,347)(123,354)(124,353)(125,352)(126,351)(127,358)(128,357)(129,356)
(130,355)(131,362)(132,361)(133,360)(134,359)(135,366)(136,365)(137,364)
(138,363)(139,370)(140,369)(141,368)(142,367)(143,374)(144,373)(145,372)
(146,371)(147,378)(148,377)(149,376)(150,375)(151,382)(152,381)(153,380)
(154,379)(155,386)(156,385)(157,384)(158,383)(159,390)(160,389)(161,388)
(162,387)(163,394)(164,393)(165,392)(166,391)(167,398)(168,397)(169,396)
(170,395)(171,402)(172,401)(173,400)(174,399)(175,406)(176,405)(177,404)
(178,403)(179,410)(180,409)(181,408)(182,407)(183,414)(184,413)(185,412)
(186,411)(187,418)(188,417)(189,416)(190,415)(191,422)(192,421)(193,420)
(194,419)(195,426)(196,425)(197,424)(198,423)(199,430)(200,429)(201,428)
(202,427)(203,434)(204,433)(205,432)(206,431)(207,438)(208,437)(209,436)
(210,435)(211,442)(212,441)(213,440)(214,439)(215,446)(216,445)(217,444)
(218,443)(219,450)(220,449)(221,448)(222,447)(223,454)(224,453)(225,452)
(226,451)(227,458)(228,457)(229,456)(230,455);
poly := sub<Sym(458)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope