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Polytope of Type {52,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {52,18}*1872b
if this polytope has a name.
Group : SmallGroup(1872,539)
Rank : 3
Schlafli Type : {52,18}
Number of vertices, edges, etc : 52, 468, 18
Order of s0s1s2 : 117
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   3-fold quotients : {52,6}*624b
   13-fold quotients : {4,18}*144c
   26-fold quotients : {4,9}*72
   39-fold quotients : {4,6}*48b
   78-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13,147)( 14,148)
( 15,145)( 16,146)( 17,151)( 18,152)( 19,149)( 20,150)( 21,155)( 22,156)
( 23,153)( 24,154)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)
( 31,137)( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,123)( 38,124)
( 39,121)( 40,122)( 41,127)( 42,128)( 43,125)( 44,126)( 45,131)( 46,132)
( 47,129)( 48,130)( 49,111)( 50,112)( 51,109)( 52,110)( 53,115)( 54,116)
( 55,113)( 56,114)( 57,119)( 58,120)( 59,117)( 60,118)( 61, 99)( 62,100)
( 63, 97)( 64, 98)( 65,103)( 66,104)( 67,101)( 68,102)( 69,107)( 70,108)
( 71,105)( 72,106)( 73, 87)( 74, 88)( 75, 85)( 76, 86)( 77, 91)( 78, 92)
( 79, 89)( 80, 90)( 81, 95)( 82, 96)( 83, 93)( 84, 94)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,303)(170,304)(171,301)(172,302)
(173,307)(174,308)(175,305)(176,306)(177,311)(178,312)(179,309)(180,310)
(181,291)(182,292)(183,289)(184,290)(185,295)(186,296)(187,293)(188,294)
(189,299)(190,300)(191,297)(192,298)(193,279)(194,280)(195,277)(196,278)
(197,283)(198,284)(199,281)(200,282)(201,287)(202,288)(203,285)(204,286)
(205,267)(206,268)(207,265)(208,266)(209,271)(210,272)(211,269)(212,270)
(213,275)(214,276)(215,273)(216,274)(217,255)(218,256)(219,253)(220,254)
(221,259)(222,260)(223,257)(224,258)(225,263)(226,264)(227,261)(228,262)
(229,243)(230,244)(231,241)(232,242)(233,247)(234,248)(235,245)(236,246)
(237,251)(238,252)(239,249)(240,250)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,459)(326,460)(327,457)(328,458)(329,463)(330,464)
(331,461)(332,462)(333,467)(334,468)(335,465)(336,466)(337,447)(338,448)
(339,445)(340,446)(341,451)(342,452)(343,449)(344,450)(345,455)(346,456)
(347,453)(348,454)(349,435)(350,436)(351,433)(352,434)(353,439)(354,440)
(355,437)(356,438)(357,443)(358,444)(359,441)(360,442)(361,423)(362,424)
(363,421)(364,422)(365,427)(366,428)(367,425)(368,426)(369,431)(370,432)
(371,429)(372,430)(373,411)(374,412)(375,409)(376,410)(377,415)(378,416)
(379,413)(380,414)(381,419)(382,420)(383,417)(384,418)(385,399)(386,400)
(387,397)(388,398)(389,403)(390,404)(391,401)(392,402)(393,407)(394,408)
(395,405)(396,406);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25,145)( 26,146)( 27,148)( 28,147)
( 29,153)( 30,154)( 31,156)( 32,155)( 33,149)( 34,150)( 35,152)( 36,151)
( 37,133)( 38,134)( 39,136)( 40,135)( 41,141)( 42,142)( 43,144)( 44,143)
( 45,137)( 46,138)( 47,140)( 48,139)( 49,121)( 50,122)( 51,124)( 52,123)
( 53,129)( 54,130)( 55,132)( 56,131)( 57,125)( 58,126)( 59,128)( 60,127)
( 61,109)( 62,110)( 63,112)( 64,111)( 65,117)( 66,118)( 67,120)( 68,119)
( 69,113)( 70,114)( 71,116)( 72,115)( 73, 97)( 74, 98)( 75,100)( 76, 99)
( 77,105)( 78,106)( 79,108)( 80,107)( 81,101)( 82,102)( 83,104)( 84,103)
( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)(157,333)(158,334)(159,336)
(160,335)(161,329)(162,330)(163,332)(164,331)(165,325)(166,326)(167,328)
(168,327)(169,321)(170,322)(171,324)(172,323)(173,317)(174,318)(175,320)
(176,319)(177,313)(178,314)(179,316)(180,315)(181,465)(182,466)(183,468)
(184,467)(185,461)(186,462)(187,464)(188,463)(189,457)(190,458)(191,460)
(192,459)(193,453)(194,454)(195,456)(196,455)(197,449)(198,450)(199,452)
(200,451)(201,445)(202,446)(203,448)(204,447)(205,441)(206,442)(207,444)
(208,443)(209,437)(210,438)(211,440)(212,439)(213,433)(214,434)(215,436)
(216,435)(217,429)(218,430)(219,432)(220,431)(221,425)(222,426)(223,428)
(224,427)(225,421)(226,422)(227,424)(228,423)(229,417)(230,418)(231,420)
(232,419)(233,413)(234,414)(235,416)(236,415)(237,409)(238,410)(239,412)
(240,411)(241,405)(242,406)(243,408)(244,407)(245,401)(246,402)(247,404)
(248,403)(249,397)(250,398)(251,400)(252,399)(253,393)(254,394)(255,396)
(256,395)(257,389)(258,390)(259,392)(260,391)(261,385)(262,386)(263,388)
(264,387)(265,381)(266,382)(267,384)(268,383)(269,377)(270,378)(271,380)
(272,379)(273,373)(274,374)(275,376)(276,375)(277,369)(278,370)(279,372)
(280,371)(281,365)(282,366)(283,368)(284,367)(285,361)(286,362)(287,364)
(288,363)(289,357)(290,358)(291,360)(292,359)(293,353)(294,354)(295,356)
(296,355)(297,349)(298,350)(299,352)(300,351)(301,345)(302,346)(303,348)
(304,347)(305,341)(306,342)(307,344)(308,343)(309,337)(310,338)(311,340)
(312,339);;
s2 := (  1,157)(  2,160)(  3,159)(  4,158)(  5,165)(  6,168)(  7,167)(  8,166)
(  9,161)( 10,164)( 11,163)( 12,162)( 13,169)( 14,172)( 15,171)( 16,170)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,173)( 22,176)( 23,175)( 24,174)
( 25,181)( 26,184)( 27,183)( 28,182)( 29,189)( 30,192)( 31,191)( 32,190)
( 33,185)( 34,188)( 35,187)( 36,186)( 37,193)( 38,196)( 39,195)( 40,194)
( 41,201)( 42,204)( 43,203)( 44,202)( 45,197)( 46,200)( 47,199)( 48,198)
( 49,205)( 50,208)( 51,207)( 52,206)( 53,213)( 54,216)( 55,215)( 56,214)
( 57,209)( 58,212)( 59,211)( 60,210)( 61,217)( 62,220)( 63,219)( 64,218)
( 65,225)( 66,228)( 67,227)( 68,226)( 69,221)( 70,224)( 71,223)( 72,222)
( 73,229)( 74,232)( 75,231)( 76,230)( 77,237)( 78,240)( 79,239)( 80,238)
( 81,233)( 82,236)( 83,235)( 84,234)( 85,241)( 86,244)( 87,243)( 88,242)
( 89,249)( 90,252)( 91,251)( 92,250)( 93,245)( 94,248)( 95,247)( 96,246)
( 97,253)( 98,256)( 99,255)(100,254)(101,261)(102,264)(103,263)(104,262)
(105,257)(106,260)(107,259)(108,258)(109,265)(110,268)(111,267)(112,266)
(113,273)(114,276)(115,275)(116,274)(117,269)(118,272)(119,271)(120,270)
(121,277)(122,280)(123,279)(124,278)(125,285)(126,288)(127,287)(128,286)
(129,281)(130,284)(131,283)(132,282)(133,289)(134,292)(135,291)(136,290)
(137,297)(138,300)(139,299)(140,298)(141,293)(142,296)(143,295)(144,294)
(145,301)(146,304)(147,303)(148,302)(149,309)(150,312)(151,311)(152,310)
(153,305)(154,308)(155,307)(156,306)(313,321)(314,324)(315,323)(316,322)
(318,320)(325,333)(326,336)(327,335)(328,334)(330,332)(337,345)(338,348)
(339,347)(340,346)(342,344)(349,357)(350,360)(351,359)(352,358)(354,356)
(361,369)(362,372)(363,371)(364,370)(366,368)(373,381)(374,384)(375,383)
(376,382)(378,380)(385,393)(386,396)(387,395)(388,394)(390,392)(397,405)
(398,408)(399,407)(400,406)(402,404)(409,417)(410,420)(411,419)(412,418)
(414,416)(421,429)(422,432)(423,431)(424,430)(426,428)(433,441)(434,444)
(435,443)(436,442)(438,440)(445,453)(446,456)(447,455)(448,454)(450,452)
(457,465)(458,468)(459,467)(460,466)(462,464);;
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(468)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13,147)
( 14,148)( 15,145)( 16,146)( 17,151)( 18,152)( 19,149)( 20,150)( 21,155)
( 22,156)( 23,153)( 24,154)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)
( 30,140)( 31,137)( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,123)
( 38,124)( 39,121)( 40,122)( 41,127)( 42,128)( 43,125)( 44,126)( 45,131)
( 46,132)( 47,129)( 48,130)( 49,111)( 50,112)( 51,109)( 52,110)( 53,115)
( 54,116)( 55,113)( 56,114)( 57,119)( 58,120)( 59,117)( 60,118)( 61, 99)
( 62,100)( 63, 97)( 64, 98)( 65,103)( 66,104)( 67,101)( 68,102)( 69,107)
( 70,108)( 71,105)( 72,106)( 73, 87)( 74, 88)( 75, 85)( 76, 86)( 77, 91)
( 78, 92)( 79, 89)( 80, 90)( 81, 95)( 82, 96)( 83, 93)( 84, 94)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,303)(170,304)(171,301)
(172,302)(173,307)(174,308)(175,305)(176,306)(177,311)(178,312)(179,309)
(180,310)(181,291)(182,292)(183,289)(184,290)(185,295)(186,296)(187,293)
(188,294)(189,299)(190,300)(191,297)(192,298)(193,279)(194,280)(195,277)
(196,278)(197,283)(198,284)(199,281)(200,282)(201,287)(202,288)(203,285)
(204,286)(205,267)(206,268)(207,265)(208,266)(209,271)(210,272)(211,269)
(212,270)(213,275)(214,276)(215,273)(216,274)(217,255)(218,256)(219,253)
(220,254)(221,259)(222,260)(223,257)(224,258)(225,263)(226,264)(227,261)
(228,262)(229,243)(230,244)(231,241)(232,242)(233,247)(234,248)(235,245)
(236,246)(237,251)(238,252)(239,249)(240,250)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,459)(326,460)(327,457)(328,458)(329,463)
(330,464)(331,461)(332,462)(333,467)(334,468)(335,465)(336,466)(337,447)
(338,448)(339,445)(340,446)(341,451)(342,452)(343,449)(344,450)(345,455)
(346,456)(347,453)(348,454)(349,435)(350,436)(351,433)(352,434)(353,439)
(354,440)(355,437)(356,438)(357,443)(358,444)(359,441)(360,442)(361,423)
(362,424)(363,421)(364,422)(365,427)(366,428)(367,425)(368,426)(369,431)
(370,432)(371,429)(372,430)(373,411)(374,412)(375,409)(376,410)(377,415)
(378,416)(379,413)(380,414)(381,419)(382,420)(383,417)(384,418)(385,399)
(386,400)(387,397)(388,398)(389,403)(390,404)(391,401)(392,402)(393,407)
(394,408)(395,405)(396,406);
s1 := Sym(468)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)
(  8, 23)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25,145)( 26,146)( 27,148)
( 28,147)( 29,153)( 30,154)( 31,156)( 32,155)( 33,149)( 34,150)( 35,152)
( 36,151)( 37,133)( 38,134)( 39,136)( 40,135)( 41,141)( 42,142)( 43,144)
( 44,143)( 45,137)( 46,138)( 47,140)( 48,139)( 49,121)( 50,122)( 51,124)
( 52,123)( 53,129)( 54,130)( 55,132)( 56,131)( 57,125)( 58,126)( 59,128)
( 60,127)( 61,109)( 62,110)( 63,112)( 64,111)( 65,117)( 66,118)( 67,120)
( 68,119)( 69,113)( 70,114)( 71,116)( 72,115)( 73, 97)( 74, 98)( 75,100)
( 76, 99)( 77,105)( 78,106)( 79,108)( 80,107)( 81,101)( 82,102)( 83,104)
( 84,103)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)(157,333)(158,334)
(159,336)(160,335)(161,329)(162,330)(163,332)(164,331)(165,325)(166,326)
(167,328)(168,327)(169,321)(170,322)(171,324)(172,323)(173,317)(174,318)
(175,320)(176,319)(177,313)(178,314)(179,316)(180,315)(181,465)(182,466)
(183,468)(184,467)(185,461)(186,462)(187,464)(188,463)(189,457)(190,458)
(191,460)(192,459)(193,453)(194,454)(195,456)(196,455)(197,449)(198,450)
(199,452)(200,451)(201,445)(202,446)(203,448)(204,447)(205,441)(206,442)
(207,444)(208,443)(209,437)(210,438)(211,440)(212,439)(213,433)(214,434)
(215,436)(216,435)(217,429)(218,430)(219,432)(220,431)(221,425)(222,426)
(223,428)(224,427)(225,421)(226,422)(227,424)(228,423)(229,417)(230,418)
(231,420)(232,419)(233,413)(234,414)(235,416)(236,415)(237,409)(238,410)
(239,412)(240,411)(241,405)(242,406)(243,408)(244,407)(245,401)(246,402)
(247,404)(248,403)(249,397)(250,398)(251,400)(252,399)(253,393)(254,394)
(255,396)(256,395)(257,389)(258,390)(259,392)(260,391)(261,385)(262,386)
(263,388)(264,387)(265,381)(266,382)(267,384)(268,383)(269,377)(270,378)
(271,380)(272,379)(273,373)(274,374)(275,376)(276,375)(277,369)(278,370)
(279,372)(280,371)(281,365)(282,366)(283,368)(284,367)(285,361)(286,362)
(287,364)(288,363)(289,357)(290,358)(291,360)(292,359)(293,353)(294,354)
(295,356)(296,355)(297,349)(298,350)(299,352)(300,351)(301,345)(302,346)
(303,348)(304,347)(305,341)(306,342)(307,344)(308,343)(309,337)(310,338)
(311,340)(312,339);
s2 := Sym(468)!(  1,157)(  2,160)(  3,159)(  4,158)(  5,165)(  6,168)(  7,167)
(  8,166)(  9,161)( 10,164)( 11,163)( 12,162)( 13,169)( 14,172)( 15,171)
( 16,170)( 17,177)( 18,180)( 19,179)( 20,178)( 21,173)( 22,176)( 23,175)
( 24,174)( 25,181)( 26,184)( 27,183)( 28,182)( 29,189)( 30,192)( 31,191)
( 32,190)( 33,185)( 34,188)( 35,187)( 36,186)( 37,193)( 38,196)( 39,195)
( 40,194)( 41,201)( 42,204)( 43,203)( 44,202)( 45,197)( 46,200)( 47,199)
( 48,198)( 49,205)( 50,208)( 51,207)( 52,206)( 53,213)( 54,216)( 55,215)
( 56,214)( 57,209)( 58,212)( 59,211)( 60,210)( 61,217)( 62,220)( 63,219)
( 64,218)( 65,225)( 66,228)( 67,227)( 68,226)( 69,221)( 70,224)( 71,223)
( 72,222)( 73,229)( 74,232)( 75,231)( 76,230)( 77,237)( 78,240)( 79,239)
( 80,238)( 81,233)( 82,236)( 83,235)( 84,234)( 85,241)( 86,244)( 87,243)
( 88,242)( 89,249)( 90,252)( 91,251)( 92,250)( 93,245)( 94,248)( 95,247)
( 96,246)( 97,253)( 98,256)( 99,255)(100,254)(101,261)(102,264)(103,263)
(104,262)(105,257)(106,260)(107,259)(108,258)(109,265)(110,268)(111,267)
(112,266)(113,273)(114,276)(115,275)(116,274)(117,269)(118,272)(119,271)
(120,270)(121,277)(122,280)(123,279)(124,278)(125,285)(126,288)(127,287)
(128,286)(129,281)(130,284)(131,283)(132,282)(133,289)(134,292)(135,291)
(136,290)(137,297)(138,300)(139,299)(140,298)(141,293)(142,296)(143,295)
(144,294)(145,301)(146,304)(147,303)(148,302)(149,309)(150,312)(151,311)
(152,310)(153,305)(154,308)(155,307)(156,306)(313,321)(314,324)(315,323)
(316,322)(318,320)(325,333)(326,336)(327,335)(328,334)(330,332)(337,345)
(338,348)(339,347)(340,346)(342,344)(349,357)(350,360)(351,359)(352,358)
(354,356)(361,369)(362,372)(363,371)(364,370)(366,368)(373,381)(374,384)
(375,383)(376,382)(378,380)(385,393)(386,396)(387,395)(388,394)(390,392)
(397,405)(398,408)(399,407)(400,406)(402,404)(409,417)(410,420)(411,419)
(412,418)(414,416)(421,429)(422,432)(423,431)(424,430)(426,428)(433,441)
(434,444)(435,443)(436,442)(438,440)(445,453)(446,456)(447,455)(448,454)
(450,452)(457,465)(458,468)(459,467)(460,466)(462,464);
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 >; 
 
References : None.
to this polytope