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Polytope of Type {18,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,54}*1944b
if this polytope has a name.
Group : SmallGroup(1944,947)
Rank : 3
Schlafli Type : {18,54}
Number of vertices, edges, etc : 18, 486, 54
Order of s0s1s2 : 54
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {18,27}*972
   3-fold quotients : {18,18}*648b, {6,54}*648b
   6-fold quotients : {18,9}*324, {6,27}*324
   9-fold quotients : {2,54}*216, {6,18}*216b
   18-fold quotients : {2,27}*108, {6,9}*108
   27-fold quotients : {2,18}*72, {6,6}*72b
   54-fold quotients : {2,9}*36, {6,3}*36
   81-fold quotients : {2,6}*24
   162-fold quotients : {2,3}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)
( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)( 42, 69)
( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)( 50, 77)
( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)(111,144)
(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)(119,152)
(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(127,160)
(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,223)(191,224)(192,225)(193,220)(194,221)(195,222)(196,217)
(197,218)(198,219)(199,232)(200,233)(201,234)(202,229)(203,230)(204,231)
(205,226)(206,227)(207,228)(208,241)(209,242)(210,243)(211,238)(212,239)
(213,240)(214,235)(215,236)(216,237)(247,250)(248,251)(249,252)(256,259)
(257,260)(258,261)(265,268)(266,269)(267,270)(271,304)(272,305)(273,306)
(274,301)(275,302)(276,303)(277,298)(278,299)(279,300)(280,313)(281,314)
(282,315)(283,310)(284,311)(285,312)(286,307)(287,308)(288,309)(289,322)
(290,323)(291,324)(292,319)(293,320)(294,321)(295,316)(296,317)(297,318)
(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)(347,350)
(348,351)(352,385)(353,386)(354,387)(355,382)(356,383)(357,384)(358,379)
(359,380)(360,381)(361,394)(362,395)(363,396)(364,391)(365,392)(366,393)
(367,388)(368,389)(369,390)(370,403)(371,404)(372,405)(373,400)(374,401)
(375,402)(376,397)(377,398)(378,399)(409,412)(410,413)(411,414)(418,421)
(419,422)(420,423)(427,430)(428,431)(429,432)(433,466)(434,467)(435,468)
(436,463)(437,464)(438,465)(439,460)(440,461)(441,462)(442,475)(443,476)
(444,477)(445,472)(446,473)(447,474)(448,469)(449,470)(450,471)(451,484)
(452,485)(453,486)(454,481)(455,482)(456,483)(457,478)(458,479)(459,480);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)( 16, 51)
( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 55, 61)( 56, 63)( 57, 62)( 59, 60)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 82,210)( 83,209)( 84,208)( 85,216)( 86,215)( 87,214)( 88,213)( 89,212)
( 90,211)( 91,201)( 92,200)( 93,199)( 94,207)( 95,206)( 96,205)( 97,204)
( 98,203)( 99,202)(100,192)(101,191)(102,190)(103,198)(104,197)(105,196)
(106,195)(107,194)(108,193)(109,183)(110,182)(111,181)(112,189)(113,188)
(114,187)(115,186)(116,185)(117,184)(118,174)(119,173)(120,172)(121,180)
(122,179)(123,178)(124,177)(125,176)(126,175)(127,165)(128,164)(129,163)
(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,243)(137,242)
(138,241)(139,240)(140,239)(141,238)(142,237)(143,236)(144,235)(145,234)
(146,233)(147,232)(148,231)(149,230)(150,229)(151,228)(152,227)(153,226)
(154,225)(155,224)(156,223)(157,222)(158,221)(159,220)(160,219)(161,218)
(162,217)(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)(250,274)
(251,276)(252,275)(253,291)(254,290)(255,289)(256,297)(257,296)(258,295)
(259,294)(260,293)(261,292)(262,282)(263,281)(264,280)(265,288)(266,287)
(267,286)(268,285)(269,284)(270,283)(298,304)(299,306)(300,305)(302,303)
(307,324)(308,323)(309,322)(310,321)(311,320)(312,319)(313,318)(314,317)
(315,316)(325,453)(326,452)(327,451)(328,459)(329,458)(330,457)(331,456)
(332,455)(333,454)(334,444)(335,443)(336,442)(337,450)(338,449)(339,448)
(340,447)(341,446)(342,445)(343,435)(344,434)(345,433)(346,441)(347,440)
(348,439)(349,438)(350,437)(351,436)(352,426)(353,425)(354,424)(355,432)
(356,431)(357,430)(358,429)(359,428)(360,427)(361,417)(362,416)(363,415)
(364,423)(365,422)(366,421)(367,420)(368,419)(369,418)(370,408)(371,407)
(372,406)(373,414)(374,413)(375,412)(376,411)(377,410)(378,409)(379,486)
(380,485)(381,484)(382,483)(383,482)(384,481)(385,480)(386,479)(387,478)
(388,477)(389,476)(390,475)(391,474)(392,473)(393,472)(394,471)(395,470)
(396,469)(397,468)(398,467)(399,466)(400,465)(401,464)(402,463)(403,462)
(404,461)(405,460);;
s2 := (  1,325)(  2,327)(  3,326)(  4,331)(  5,333)(  6,332)(  7,328)(  8,330)
(  9,329)( 10,345)( 11,344)( 12,343)( 13,351)( 14,350)( 15,349)( 16,348)
( 17,347)( 18,346)( 19,336)( 20,335)( 21,334)( 22,342)( 23,341)( 24,340)
( 25,339)( 26,338)( 27,337)( 28,385)( 29,387)( 30,386)( 31,382)( 32,384)
( 33,383)( 34,379)( 35,381)( 36,380)( 37,405)( 38,404)( 39,403)( 40,402)
( 41,401)( 42,400)( 43,399)( 44,398)( 45,397)( 46,396)( 47,395)( 48,394)
( 49,393)( 50,392)( 51,391)( 52,390)( 53,389)( 54,388)( 55,358)( 56,360)
( 57,359)( 58,355)( 59,357)( 60,356)( 61,352)( 62,354)( 63,353)( 64,378)
( 65,377)( 66,376)( 67,375)( 68,374)( 69,373)( 70,372)( 71,371)( 72,370)
( 73,369)( 74,368)( 75,367)( 76,366)( 77,365)( 78,364)( 79,363)( 80,362)
( 81,361)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)( 88,247)
( 89,249)( 90,248)( 91,264)( 92,263)( 93,262)( 94,270)( 95,269)( 96,268)
( 97,267)( 98,266)( 99,265)(100,255)(101,254)(102,253)(103,261)(104,260)
(105,259)(106,258)(107,257)(108,256)(109,304)(110,306)(111,305)(112,301)
(113,303)(114,302)(115,298)(116,300)(117,299)(118,324)(119,323)(120,322)
(121,321)(122,320)(123,319)(124,318)(125,317)(126,316)(127,315)(128,314)
(129,313)(130,312)(131,311)(132,310)(133,309)(134,308)(135,307)(136,277)
(137,279)(138,278)(139,274)(140,276)(141,275)(142,271)(143,273)(144,272)
(145,297)(146,296)(147,295)(148,294)(149,293)(150,292)(151,291)(152,290)
(153,289)(154,288)(155,287)(156,286)(157,285)(158,284)(159,283)(160,282)
(161,281)(162,280)(163,426)(164,425)(165,424)(166,432)(167,431)(168,430)
(169,429)(170,428)(171,427)(172,417)(173,416)(174,415)(175,423)(176,422)
(177,421)(178,420)(179,419)(180,418)(181,408)(182,407)(183,406)(184,414)
(185,413)(186,412)(187,411)(188,410)(189,409)(190,486)(191,485)(192,484)
(193,483)(194,482)(195,481)(196,480)(197,479)(198,478)(199,477)(200,476)
(201,475)(202,474)(203,473)(204,472)(205,471)(206,470)(207,469)(208,468)
(209,467)(210,466)(211,465)(212,464)(213,463)(214,462)(215,461)(216,460)
(217,459)(218,458)(219,457)(220,456)(221,455)(222,454)(223,453)(224,452)
(225,451)(226,450)(227,449)(228,448)(229,447)(230,446)(231,445)(232,444)
(233,443)(234,442)(235,441)(236,440)(237,439)(238,438)(239,437)(240,436)
(241,435)(242,434)(243,433);;
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, s2*s0*s1*s0*s1*s2*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, 
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(486)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)
( 34, 55)( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)
( 42, 69)( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)
( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)
(111,144)(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)
(119,152)(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)
(127,160)(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)
(135,156)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,223)(191,224)(192,225)(193,220)(194,221)(195,222)
(196,217)(197,218)(198,219)(199,232)(200,233)(201,234)(202,229)(203,230)
(204,231)(205,226)(206,227)(207,228)(208,241)(209,242)(210,243)(211,238)
(212,239)(213,240)(214,235)(215,236)(216,237)(247,250)(248,251)(249,252)
(256,259)(257,260)(258,261)(265,268)(266,269)(267,270)(271,304)(272,305)
(273,306)(274,301)(275,302)(276,303)(277,298)(278,299)(279,300)(280,313)
(281,314)(282,315)(283,310)(284,311)(285,312)(286,307)(287,308)(288,309)
(289,322)(290,323)(291,324)(292,319)(293,320)(294,321)(295,316)(296,317)
(297,318)(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)
(347,350)(348,351)(352,385)(353,386)(354,387)(355,382)(356,383)(357,384)
(358,379)(359,380)(360,381)(361,394)(362,395)(363,396)(364,391)(365,392)
(366,393)(367,388)(368,389)(369,390)(370,403)(371,404)(372,405)(373,400)
(374,401)(375,402)(376,397)(377,398)(378,399)(409,412)(410,413)(411,414)
(418,421)(419,422)(420,423)(427,430)(428,431)(429,432)(433,466)(434,467)
(435,468)(436,463)(437,464)(438,465)(439,460)(440,461)(441,462)(442,475)
(443,476)(444,477)(445,472)(446,473)(447,474)(448,469)(449,470)(450,471)
(451,484)(452,485)(453,486)(454,481)(455,482)(456,483)(457,478)(458,479)
(459,480);
s1 := Sym(486)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)
( 16, 51)( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 61)( 56, 63)( 57, 62)( 59, 60)
( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)
( 72, 73)( 82,210)( 83,209)( 84,208)( 85,216)( 86,215)( 87,214)( 88,213)
( 89,212)( 90,211)( 91,201)( 92,200)( 93,199)( 94,207)( 95,206)( 96,205)
( 97,204)( 98,203)( 99,202)(100,192)(101,191)(102,190)(103,198)(104,197)
(105,196)(106,195)(107,194)(108,193)(109,183)(110,182)(111,181)(112,189)
(113,188)(114,187)(115,186)(116,185)(117,184)(118,174)(119,173)(120,172)
(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,165)(128,164)
(129,163)(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,243)
(137,242)(138,241)(139,240)(140,239)(141,238)(142,237)(143,236)(144,235)
(145,234)(146,233)(147,232)(148,231)(149,230)(150,229)(151,228)(152,227)
(153,226)(154,225)(155,224)(156,223)(157,222)(158,221)(159,220)(160,219)
(161,218)(162,217)(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)
(250,274)(251,276)(252,275)(253,291)(254,290)(255,289)(256,297)(257,296)
(258,295)(259,294)(260,293)(261,292)(262,282)(263,281)(264,280)(265,288)
(266,287)(267,286)(268,285)(269,284)(270,283)(298,304)(299,306)(300,305)
(302,303)(307,324)(308,323)(309,322)(310,321)(311,320)(312,319)(313,318)
(314,317)(315,316)(325,453)(326,452)(327,451)(328,459)(329,458)(330,457)
(331,456)(332,455)(333,454)(334,444)(335,443)(336,442)(337,450)(338,449)
(339,448)(340,447)(341,446)(342,445)(343,435)(344,434)(345,433)(346,441)
(347,440)(348,439)(349,438)(350,437)(351,436)(352,426)(353,425)(354,424)
(355,432)(356,431)(357,430)(358,429)(359,428)(360,427)(361,417)(362,416)
(363,415)(364,423)(365,422)(366,421)(367,420)(368,419)(369,418)(370,408)
(371,407)(372,406)(373,414)(374,413)(375,412)(376,411)(377,410)(378,409)
(379,486)(380,485)(381,484)(382,483)(383,482)(384,481)(385,480)(386,479)
(387,478)(388,477)(389,476)(390,475)(391,474)(392,473)(393,472)(394,471)
(395,470)(396,469)(397,468)(398,467)(399,466)(400,465)(401,464)(402,463)
(403,462)(404,461)(405,460);
s2 := Sym(486)!(  1,325)(  2,327)(  3,326)(  4,331)(  5,333)(  6,332)(  7,328)
(  8,330)(  9,329)( 10,345)( 11,344)( 12,343)( 13,351)( 14,350)( 15,349)
( 16,348)( 17,347)( 18,346)( 19,336)( 20,335)( 21,334)( 22,342)( 23,341)
( 24,340)( 25,339)( 26,338)( 27,337)( 28,385)( 29,387)( 30,386)( 31,382)
( 32,384)( 33,383)( 34,379)( 35,381)( 36,380)( 37,405)( 38,404)( 39,403)
( 40,402)( 41,401)( 42,400)( 43,399)( 44,398)( 45,397)( 46,396)( 47,395)
( 48,394)( 49,393)( 50,392)( 51,391)( 52,390)( 53,389)( 54,388)( 55,358)
( 56,360)( 57,359)( 58,355)( 59,357)( 60,356)( 61,352)( 62,354)( 63,353)
( 64,378)( 65,377)( 66,376)( 67,375)( 68,374)( 69,373)( 70,372)( 71,371)
( 72,370)( 73,369)( 74,368)( 75,367)( 76,366)( 77,365)( 78,364)( 79,363)
( 80,362)( 81,361)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)
( 88,247)( 89,249)( 90,248)( 91,264)( 92,263)( 93,262)( 94,270)( 95,269)
( 96,268)( 97,267)( 98,266)( 99,265)(100,255)(101,254)(102,253)(103,261)
(104,260)(105,259)(106,258)(107,257)(108,256)(109,304)(110,306)(111,305)
(112,301)(113,303)(114,302)(115,298)(116,300)(117,299)(118,324)(119,323)
(120,322)(121,321)(122,320)(123,319)(124,318)(125,317)(126,316)(127,315)
(128,314)(129,313)(130,312)(131,311)(132,310)(133,309)(134,308)(135,307)
(136,277)(137,279)(138,278)(139,274)(140,276)(141,275)(142,271)(143,273)
(144,272)(145,297)(146,296)(147,295)(148,294)(149,293)(150,292)(151,291)
(152,290)(153,289)(154,288)(155,287)(156,286)(157,285)(158,284)(159,283)
(160,282)(161,281)(162,280)(163,426)(164,425)(165,424)(166,432)(167,431)
(168,430)(169,429)(170,428)(171,427)(172,417)(173,416)(174,415)(175,423)
(176,422)(177,421)(178,420)(179,419)(180,418)(181,408)(182,407)(183,406)
(184,414)(185,413)(186,412)(187,411)(188,410)(189,409)(190,486)(191,485)
(192,484)(193,483)(194,482)(195,481)(196,480)(197,479)(198,478)(199,477)
(200,476)(201,475)(202,474)(203,473)(204,472)(205,471)(206,470)(207,469)
(208,468)(209,467)(210,466)(211,465)(212,464)(213,463)(214,462)(215,461)
(216,460)(217,459)(218,458)(219,457)(220,456)(221,455)(222,454)(223,453)
(224,452)(225,451)(226,450)(227,449)(228,448)(229,447)(230,446)(231,445)
(232,444)(233,443)(234,442)(235,441)(236,440)(237,439)(238,438)(239,437)
(240,436)(241,435)(242,434)(243,433);
poly := sub<Sym(486)|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, s2*s0*s1*s0*s1*s2*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, 
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