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