Polytope of Type {18,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,54}*1944a
Also Known As : {18,54|2}. if this polytope has another 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 : 2
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) :
3-fold quotients : {18,18}*648a, {6,54}*648a
9-fold quotients : {2,54}*216, {6,18}*216a, {18,6}*216a
18-fold quotients : {2,27}*108
27-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a
54-fold quotients : {2,9}*36, {9,2}*36
81-fold quotients : {2,6}*24, {6,2}*24
162-fold quotients : {2,3}*12, {3,2}*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)( 4,328)( 5,330)( 6,329)( 7,331)( 8,333)
( 9,332)( 10,345)( 11,344)( 12,343)( 13,348)( 14,347)( 15,346)( 16,351)
( 17,350)( 18,349)( 19,336)( 20,335)( 21,334)( 22,339)( 23,338)( 24,337)
( 25,342)( 26,341)( 27,340)( 28,352)( 29,354)( 30,353)( 31,355)( 32,357)
( 33,356)( 34,358)( 35,360)( 36,359)( 37,372)( 38,371)( 39,370)( 40,375)
( 41,374)( 42,373)( 43,378)( 44,377)( 45,376)( 46,363)( 47,362)( 48,361)
( 49,366)( 50,365)( 51,364)( 52,369)( 53,368)( 54,367)( 55,379)( 56,381)
( 57,380)( 58,382)( 59,384)( 60,383)( 61,385)( 62,387)( 63,386)( 64,399)
( 65,398)( 66,397)( 67,402)( 68,401)( 69,400)( 70,405)( 71,404)( 72,403)
( 73,390)( 74,389)( 75,388)( 76,393)( 77,392)( 78,391)( 79,396)( 80,395)
( 81,394)( 82,244)( 83,246)( 84,245)( 85,247)( 86,249)( 87,248)( 88,250)
( 89,252)( 90,251)( 91,264)( 92,263)( 93,262)( 94,267)( 95,266)( 96,265)
( 97,270)( 98,269)( 99,268)(100,255)(101,254)(102,253)(103,258)(104,257)
(105,256)(106,261)(107,260)(108,259)(109,271)(110,273)(111,272)(112,274)
(113,276)(114,275)(115,277)(116,279)(117,278)(118,291)(119,290)(120,289)
(121,294)(122,293)(123,292)(124,297)(125,296)(126,295)(127,282)(128,281)
(129,280)(130,285)(131,284)(132,283)(133,288)(134,287)(135,286)(136,298)
(137,300)(138,299)(139,301)(140,303)(141,302)(142,304)(143,306)(144,305)
(145,318)(146,317)(147,316)(148,321)(149,320)(150,319)(151,324)(152,323)
(153,322)(154,309)(155,308)(156,307)(157,312)(158,311)(159,310)(160,315)
(161,314)(162,313)(163,426)(164,425)(165,424)(166,429)(167,428)(168,427)
(169,432)(170,431)(171,430)(172,417)(173,416)(174,415)(175,420)(176,419)
(177,418)(178,423)(179,422)(180,421)(181,408)(182,407)(183,406)(184,411)
(185,410)(186,409)(187,414)(188,413)(189,412)(190,453)(191,452)(192,451)
(193,456)(194,455)(195,454)(196,459)(197,458)(198,457)(199,444)(200,443)
(201,442)(202,447)(203,446)(204,445)(205,450)(206,449)(207,448)(208,435)
(209,434)(210,433)(211,438)(212,437)(213,436)(214,441)(215,440)(216,439)
(217,480)(218,479)(219,478)(220,483)(221,482)(222,481)(223,486)(224,485)
(225,484)(226,471)(227,470)(228,469)(229,474)(230,473)(231,472)(232,477)
(233,476)(234,475)(235,462)(236,461)(237,460)(238,465)(239,464)(240,463)
(241,468)(242,467)(243,466);;
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*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)( 4,328)( 5,330)( 6,329)( 7,331)
( 8,333)( 9,332)( 10,345)( 11,344)( 12,343)( 13,348)( 14,347)( 15,346)
( 16,351)( 17,350)( 18,349)( 19,336)( 20,335)( 21,334)( 22,339)( 23,338)
( 24,337)( 25,342)( 26,341)( 27,340)( 28,352)( 29,354)( 30,353)( 31,355)
( 32,357)( 33,356)( 34,358)( 35,360)( 36,359)( 37,372)( 38,371)( 39,370)
( 40,375)( 41,374)( 42,373)( 43,378)( 44,377)( 45,376)( 46,363)( 47,362)
( 48,361)( 49,366)( 50,365)( 51,364)( 52,369)( 53,368)( 54,367)( 55,379)
( 56,381)( 57,380)( 58,382)( 59,384)( 60,383)( 61,385)( 62,387)( 63,386)
( 64,399)( 65,398)( 66,397)( 67,402)( 68,401)( 69,400)( 70,405)( 71,404)
( 72,403)( 73,390)( 74,389)( 75,388)( 76,393)( 77,392)( 78,391)( 79,396)
( 80,395)( 81,394)( 82,244)( 83,246)( 84,245)( 85,247)( 86,249)( 87,248)
( 88,250)( 89,252)( 90,251)( 91,264)( 92,263)( 93,262)( 94,267)( 95,266)
( 96,265)( 97,270)( 98,269)( 99,268)(100,255)(101,254)(102,253)(103,258)
(104,257)(105,256)(106,261)(107,260)(108,259)(109,271)(110,273)(111,272)
(112,274)(113,276)(114,275)(115,277)(116,279)(117,278)(118,291)(119,290)
(120,289)(121,294)(122,293)(123,292)(124,297)(125,296)(126,295)(127,282)
(128,281)(129,280)(130,285)(131,284)(132,283)(133,288)(134,287)(135,286)
(136,298)(137,300)(138,299)(139,301)(140,303)(141,302)(142,304)(143,306)
(144,305)(145,318)(146,317)(147,316)(148,321)(149,320)(150,319)(151,324)
(152,323)(153,322)(154,309)(155,308)(156,307)(157,312)(158,311)(159,310)
(160,315)(161,314)(162,313)(163,426)(164,425)(165,424)(166,429)(167,428)
(168,427)(169,432)(170,431)(171,430)(172,417)(173,416)(174,415)(175,420)
(176,419)(177,418)(178,423)(179,422)(180,421)(181,408)(182,407)(183,406)
(184,411)(185,410)(186,409)(187,414)(188,413)(189,412)(190,453)(191,452)
(192,451)(193,456)(194,455)(195,454)(196,459)(197,458)(198,457)(199,444)
(200,443)(201,442)(202,447)(203,446)(204,445)(205,450)(206,449)(207,448)
(208,435)(209,434)(210,433)(211,438)(212,437)(213,436)(214,441)(215,440)
(216,439)(217,480)(218,479)(219,478)(220,483)(221,482)(222,481)(223,486)
(224,485)(225,484)(226,471)(227,470)(228,469)(229,474)(230,473)(231,472)
(232,477)(233,476)(234,475)(235,462)(236,461)(237,460)(238,465)(239,464)
(240,463)(241,468)(242,467)(243,466);
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, s0*s1*s2*s1*s0*s1*s2*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