Polytope of Type {6,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*1944f
if this polytope has a name.
Group : SmallGroup(1944,954)
Rank : 3
Schlafli Type : {6,54}
Number of vertices, edges, etc : 18, 486, 162
Order of s0s1s2 : 54
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,54}*972c
3-fold quotients : {6,18}*648b
6-fold quotients : {6,18}*324a
9-fold quotients : {6,18}*216a, {6,6}*216b
18-fold quotients : {6,6}*108
27-fold quotients : {2,18}*72, {6,6}*72a
54-fold quotients : {2,9}*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, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)( 34, 58)
( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)( 42, 72)
( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)( 50, 80)
( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)(196,220)
(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)(204,234)
(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)(212,242)
(213,243)(214,238)(215,239)(216,240)(247,250)(248,251)(249,252)(256,259)
(257,260)(258,261)(265,268)(266,269)(267,270)(271,298)(272,299)(273,300)
(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)(281,308)
(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)(289,316)
(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)(297,321)
(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)(347,350)
(348,351)(352,379)(353,380)(354,381)(355,385)(356,386)(357,387)(358,382)
(359,383)(360,384)(361,388)(362,389)(363,390)(364,394)(365,395)(366,396)
(367,391)(368,392)(369,393)(370,397)(371,398)(372,399)(373,403)(374,404)
(375,405)(376,400)(377,401)(378,402)(409,412)(410,413)(411,414)(418,421)
(419,422)(420,423)(427,430)(428,431)(429,432)(433,460)(434,461)(435,462)
(436,466)(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)(443,470)
(444,471)(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)(451,478)
(452,479)(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)(459,483);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 33)( 5, 32)( 6, 31)( 7, 35)( 8, 34)
( 9, 36)( 10, 47)( 11, 46)( 12, 48)( 13, 49)( 14, 51)( 15, 50)( 16, 54)
( 17, 53)( 18, 52)( 19, 38)( 20, 37)( 21, 39)( 22, 40)( 23, 42)( 24, 41)
( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 58, 60)( 61, 62)( 64, 74)( 65, 73)
( 66, 75)( 67, 76)( 68, 78)( 69, 77)( 70, 81)( 71, 80)( 72, 79)( 82,200)
( 83,199)( 84,201)( 85,202)( 86,204)( 87,203)( 88,207)( 89,206)( 90,205)
( 91,191)( 92,190)( 93,192)( 94,193)( 95,195)( 96,194)( 97,198)( 98,197)
( 99,196)(100,210)(101,209)(102,208)(103,212)(104,211)(105,213)(106,214)
(107,216)(108,215)(109,173)(110,172)(111,174)(112,175)(113,177)(114,176)
(115,180)(116,179)(117,178)(118,164)(119,163)(120,165)(121,166)(122,168)
(123,167)(124,171)(125,170)(126,169)(127,183)(128,182)(129,181)(130,185)
(131,184)(132,186)(133,187)(134,189)(135,188)(136,227)(137,226)(138,228)
(139,229)(140,231)(141,230)(142,234)(143,233)(144,232)(145,218)(146,217)
(147,219)(148,220)(149,222)(150,221)(151,225)(152,224)(153,223)(154,237)
(155,236)(156,235)(157,239)(158,238)(159,240)(160,241)(161,243)(162,242)
(244,271)(245,273)(246,272)(247,276)(248,275)(249,274)(250,278)(251,277)
(252,279)(253,290)(254,289)(255,291)(256,292)(257,294)(258,293)(259,297)
(260,296)(261,295)(262,281)(263,280)(264,282)(265,283)(266,285)(267,284)
(268,288)(269,287)(270,286)(299,300)(301,303)(304,305)(307,317)(308,316)
(309,318)(310,319)(311,321)(312,320)(313,324)(314,323)(315,322)(325,443)
(326,442)(327,444)(328,445)(329,447)(330,446)(331,450)(332,449)(333,448)
(334,434)(335,433)(336,435)(337,436)(338,438)(339,437)(340,441)(341,440)
(342,439)(343,453)(344,452)(345,451)(346,455)(347,454)(348,456)(349,457)
(350,459)(351,458)(352,416)(353,415)(354,417)(355,418)(356,420)(357,419)
(358,423)(359,422)(360,421)(361,407)(362,406)(363,408)(364,409)(365,411)
(366,410)(367,414)(368,413)(369,412)(370,426)(371,425)(372,424)(373,428)
(374,427)(375,429)(376,430)(377,432)(378,431)(379,470)(380,469)(381,471)
(382,472)(383,474)(384,473)(385,477)(386,476)(387,475)(388,461)(389,460)
(390,462)(391,463)(392,465)(393,464)(394,468)(395,467)(396,466)(397,480)
(398,479)(399,478)(400,482)(401,481)(402,483)(403,484)(404,486)(405,485);;
s2 := ( 1,325)( 2,327)( 3,326)( 4,331)( 5,333)( 6,332)( 7,328)( 8,330)
( 9,329)( 10,344)( 11,343)( 12,345)( 13,350)( 14,349)( 15,351)( 16,347)
( 17,346)( 18,348)( 19,335)( 20,334)( 21,336)( 22,341)( 23,340)( 24,342)
( 25,338)( 26,337)( 27,339)( 28,356)( 29,355)( 30,357)( 31,353)( 32,352)
( 33,354)( 34,359)( 35,358)( 36,360)( 37,375)( 38,374)( 39,373)( 40,372)
( 41,371)( 42,370)( 43,378)( 44,377)( 45,376)( 46,366)( 47,365)( 48,364)
( 49,363)( 50,362)( 51,361)( 52,369)( 53,368)( 54,367)( 55,386)( 56,385)
( 57,387)( 58,383)( 59,382)( 60,384)( 61,380)( 62,379)( 63,381)( 64,405)
( 65,404)( 66,403)( 67,402)( 68,401)( 69,400)( 70,399)( 71,398)( 72,397)
( 73,396)( 74,395)( 75,394)( 76,393)( 77,392)( 78,391)( 79,390)( 80,389)
( 81,388)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)( 88,247)
( 89,249)( 90,248)( 91,263)( 92,262)( 93,264)( 94,269)( 95,268)( 96,270)
( 97,266)( 98,265)( 99,267)(100,254)(101,253)(102,255)(103,260)(104,259)
(105,261)(106,257)(107,256)(108,258)(109,275)(110,274)(111,276)(112,272)
(113,271)(114,273)(115,278)(116,277)(117,279)(118,294)(119,293)(120,292)
(121,291)(122,290)(123,289)(124,297)(125,296)(126,295)(127,285)(128,284)
(129,283)(130,282)(131,281)(132,280)(133,288)(134,287)(135,286)(136,305)
(137,304)(138,306)(139,302)(140,301)(141,303)(142,299)(143,298)(144,300)
(145,324)(146,323)(147,322)(148,321)(149,320)(150,319)(151,318)(152,317)
(153,316)(154,315)(155,314)(156,313)(157,312)(158,311)(159,310)(160,309)
(161,308)(162,307)(163,416)(164,415)(165,417)(166,422)(167,421)(168,423)
(169,419)(170,418)(171,420)(172,407)(173,406)(174,408)(175,413)(176,412)
(177,414)(178,410)(179,409)(180,411)(181,426)(182,425)(183,424)(184,432)
(185,431)(186,430)(187,429)(188,428)(189,427)(190,447)(191,446)(192,445)
(193,444)(194,443)(195,442)(196,450)(197,449)(198,448)(199,438)(200,437)
(201,436)(202,435)(203,434)(204,433)(205,441)(206,440)(207,439)(208,454)
(209,456)(210,455)(211,451)(212,453)(213,452)(214,457)(215,459)(216,458)
(217,477)(218,476)(219,475)(220,474)(221,473)(222,472)(223,471)(224,470)
(225,469)(226,468)(227,467)(228,466)(229,465)(230,464)(231,463)(232,462)
(233,461)(234,460)(235,484)(236,486)(237,485)(238,481)(239,483)(240,482)
(241,478)(242,480)(243,479);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 ];;
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, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)
( 34, 58)( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)
( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)
( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)
(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)
(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)
(135,159)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)
(196,220)(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)
(204,234)(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)
(212,242)(213,243)(214,238)(215,239)(216,240)(247,250)(248,251)(249,252)
(256,259)(257,260)(258,261)(265,268)(266,269)(267,270)(271,298)(272,299)
(273,300)(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)
(281,308)(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)
(289,316)(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)
(297,321)(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)
(347,350)(348,351)(352,379)(353,380)(354,381)(355,385)(356,386)(357,387)
(358,382)(359,383)(360,384)(361,388)(362,389)(363,390)(364,394)(365,395)
(366,396)(367,391)(368,392)(369,393)(370,397)(371,398)(372,399)(373,403)
(374,404)(375,405)(376,400)(377,401)(378,402)(409,412)(410,413)(411,414)
(418,421)(419,422)(420,423)(427,430)(428,431)(429,432)(433,460)(434,461)
(435,462)(436,466)(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)
(443,470)(444,471)(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)
(451,478)(452,479)(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)
(459,483);
s1 := Sym(486)!( 1, 28)( 2, 30)( 3, 29)( 4, 33)( 5, 32)( 6, 31)( 7, 35)
( 8, 34)( 9, 36)( 10, 47)( 11, 46)( 12, 48)( 13, 49)( 14, 51)( 15, 50)
( 16, 54)( 17, 53)( 18, 52)( 19, 38)( 20, 37)( 21, 39)( 22, 40)( 23, 42)
( 24, 41)( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 58, 60)( 61, 62)( 64, 74)
( 65, 73)( 66, 75)( 67, 76)( 68, 78)( 69, 77)( 70, 81)( 71, 80)( 72, 79)
( 82,200)( 83,199)( 84,201)( 85,202)( 86,204)( 87,203)( 88,207)( 89,206)
( 90,205)( 91,191)( 92,190)( 93,192)( 94,193)( 95,195)( 96,194)( 97,198)
( 98,197)( 99,196)(100,210)(101,209)(102,208)(103,212)(104,211)(105,213)
(106,214)(107,216)(108,215)(109,173)(110,172)(111,174)(112,175)(113,177)
(114,176)(115,180)(116,179)(117,178)(118,164)(119,163)(120,165)(121,166)
(122,168)(123,167)(124,171)(125,170)(126,169)(127,183)(128,182)(129,181)
(130,185)(131,184)(132,186)(133,187)(134,189)(135,188)(136,227)(137,226)
(138,228)(139,229)(140,231)(141,230)(142,234)(143,233)(144,232)(145,218)
(146,217)(147,219)(148,220)(149,222)(150,221)(151,225)(152,224)(153,223)
(154,237)(155,236)(156,235)(157,239)(158,238)(159,240)(160,241)(161,243)
(162,242)(244,271)(245,273)(246,272)(247,276)(248,275)(249,274)(250,278)
(251,277)(252,279)(253,290)(254,289)(255,291)(256,292)(257,294)(258,293)
(259,297)(260,296)(261,295)(262,281)(263,280)(264,282)(265,283)(266,285)
(267,284)(268,288)(269,287)(270,286)(299,300)(301,303)(304,305)(307,317)
(308,316)(309,318)(310,319)(311,321)(312,320)(313,324)(314,323)(315,322)
(325,443)(326,442)(327,444)(328,445)(329,447)(330,446)(331,450)(332,449)
(333,448)(334,434)(335,433)(336,435)(337,436)(338,438)(339,437)(340,441)
(341,440)(342,439)(343,453)(344,452)(345,451)(346,455)(347,454)(348,456)
(349,457)(350,459)(351,458)(352,416)(353,415)(354,417)(355,418)(356,420)
(357,419)(358,423)(359,422)(360,421)(361,407)(362,406)(363,408)(364,409)
(365,411)(366,410)(367,414)(368,413)(369,412)(370,426)(371,425)(372,424)
(373,428)(374,427)(375,429)(376,430)(377,432)(378,431)(379,470)(380,469)
(381,471)(382,472)(383,474)(384,473)(385,477)(386,476)(387,475)(388,461)
(389,460)(390,462)(391,463)(392,465)(393,464)(394,468)(395,467)(396,466)
(397,480)(398,479)(399,478)(400,482)(401,481)(402,483)(403,484)(404,486)
(405,485);
s2 := Sym(486)!( 1,325)( 2,327)( 3,326)( 4,331)( 5,333)( 6,332)( 7,328)
( 8,330)( 9,329)( 10,344)( 11,343)( 12,345)( 13,350)( 14,349)( 15,351)
( 16,347)( 17,346)( 18,348)( 19,335)( 20,334)( 21,336)( 22,341)( 23,340)
( 24,342)( 25,338)( 26,337)( 27,339)( 28,356)( 29,355)( 30,357)( 31,353)
( 32,352)( 33,354)( 34,359)( 35,358)( 36,360)( 37,375)( 38,374)( 39,373)
( 40,372)( 41,371)( 42,370)( 43,378)( 44,377)( 45,376)( 46,366)( 47,365)
( 48,364)( 49,363)( 50,362)( 51,361)( 52,369)( 53,368)( 54,367)( 55,386)
( 56,385)( 57,387)( 58,383)( 59,382)( 60,384)( 61,380)( 62,379)( 63,381)
( 64,405)( 65,404)( 66,403)( 67,402)( 68,401)( 69,400)( 70,399)( 71,398)
( 72,397)( 73,396)( 74,395)( 75,394)( 76,393)( 77,392)( 78,391)( 79,390)
( 80,389)( 81,388)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)
( 88,247)( 89,249)( 90,248)( 91,263)( 92,262)( 93,264)( 94,269)( 95,268)
( 96,270)( 97,266)( 98,265)( 99,267)(100,254)(101,253)(102,255)(103,260)
(104,259)(105,261)(106,257)(107,256)(108,258)(109,275)(110,274)(111,276)
(112,272)(113,271)(114,273)(115,278)(116,277)(117,279)(118,294)(119,293)
(120,292)(121,291)(122,290)(123,289)(124,297)(125,296)(126,295)(127,285)
(128,284)(129,283)(130,282)(131,281)(132,280)(133,288)(134,287)(135,286)
(136,305)(137,304)(138,306)(139,302)(140,301)(141,303)(142,299)(143,298)
(144,300)(145,324)(146,323)(147,322)(148,321)(149,320)(150,319)(151,318)
(152,317)(153,316)(154,315)(155,314)(156,313)(157,312)(158,311)(159,310)
(160,309)(161,308)(162,307)(163,416)(164,415)(165,417)(166,422)(167,421)
(168,423)(169,419)(170,418)(171,420)(172,407)(173,406)(174,408)(175,413)
(176,412)(177,414)(178,410)(179,409)(180,411)(181,426)(182,425)(183,424)
(184,432)(185,431)(186,430)(187,429)(188,428)(189,427)(190,447)(191,446)
(192,445)(193,444)(194,443)(195,442)(196,450)(197,449)(198,448)(199,438)
(200,437)(201,436)(202,435)(203,434)(204,433)(205,441)(206,440)(207,439)
(208,454)(209,456)(210,455)(211,451)(212,453)(213,452)(214,457)(215,459)
(216,458)(217,477)(218,476)(219,475)(220,474)(221,473)(222,472)(223,471)
(224,470)(225,469)(226,468)(227,467)(228,466)(229,465)(230,464)(231,463)
(232,462)(233,461)(234,460)(235,484)(236,486)(237,485)(238,481)(239,483)
(240,482)(241,478)(242,480)(243,479);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope