Polytope of Type {6,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*1944d
if this polytope has a name.
Group : SmallGroup(1944,953)
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}*972b
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, 32)( 5, 31)( 6, 33)( 7, 36)( 8, 35)
( 9, 34)( 10, 47)( 11, 46)( 12, 48)( 13, 51)( 14, 50)( 15, 49)( 16, 52)
( 17, 54)( 18, 53)( 19, 38)( 20, 37)( 21, 39)( 22, 42)( 23, 41)( 24, 40)
( 25, 43)( 26, 45)( 27, 44)( 56, 57)( 58, 59)( 61, 63)( 64, 74)( 65, 73)
( 66, 75)( 67, 78)( 68, 77)( 69, 76)( 70, 79)( 71, 81)( 72, 80)( 82,200)
( 83,199)( 84,201)( 85,204)( 86,203)( 87,202)( 88,205)( 89,207)( 90,206)
( 91,191)( 92,190)( 93,192)( 94,195)( 95,194)( 96,193)( 97,196)( 98,198)
( 99,197)(100,210)(101,209)(102,208)(103,211)(104,213)(105,212)(106,215)
(107,214)(108,216)(109,173)(110,172)(111,174)(112,177)(113,176)(114,175)
(115,178)(116,180)(117,179)(118,164)(119,163)(120,165)(121,168)(122,167)
(123,166)(124,169)(125,171)(126,170)(127,183)(128,182)(129,181)(130,184)
(131,186)(132,185)(133,188)(134,187)(135,189)(136,227)(137,226)(138,228)
(139,231)(140,230)(141,229)(142,232)(143,234)(144,233)(145,218)(146,217)
(147,219)(148,222)(149,221)(150,220)(151,223)(152,225)(153,224)(154,237)
(155,236)(156,235)(157,238)(158,240)(159,239)(160,242)(161,241)(162,243)
(244,271)(245,273)(246,272)(247,275)(248,274)(249,276)(250,279)(251,278)
(252,277)(253,290)(254,289)(255,291)(256,294)(257,293)(258,292)(259,295)
(260,297)(261,296)(262,281)(263,280)(264,282)(265,285)(266,284)(267,283)
(268,286)(269,288)(270,287)(299,300)(301,302)(304,306)(307,317)(308,316)
(309,318)(310,321)(311,320)(312,319)(313,322)(314,324)(315,323)(325,443)
(326,442)(327,444)(328,447)(329,446)(330,445)(331,448)(332,450)(333,449)
(334,434)(335,433)(336,435)(337,438)(338,437)(339,436)(340,439)(341,441)
(342,440)(343,453)(344,452)(345,451)(346,454)(347,456)(348,455)(349,458)
(350,457)(351,459)(352,416)(353,415)(354,417)(355,420)(356,419)(357,418)
(358,421)(359,423)(360,422)(361,407)(362,406)(363,408)(364,411)(365,410)
(366,409)(367,412)(368,414)(369,413)(370,426)(371,425)(372,424)(373,427)
(374,429)(375,428)(376,431)(377,430)(378,432)(379,470)(380,469)(381,471)
(382,474)(383,473)(384,472)(385,475)(386,477)(387,476)(388,461)(389,460)
(390,462)(391,465)(392,464)(393,463)(394,466)(395,468)(396,467)(397,480)
(398,479)(399,478)(400,481)(401,483)(402,482)(403,485)(404,484)(405,486);;
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)( 28,357)( 29,356)( 30,355)( 31,354)( 32,353)
( 33,352)( 34,360)( 35,359)( 36,358)( 37,373)( 38,375)( 39,374)( 40,370)
( 41,372)( 42,371)( 43,376)( 44,378)( 45,377)( 46,364)( 47,366)( 48,365)
( 49,361)( 50,363)( 51,362)( 52,367)( 53,369)( 54,368)( 55,387)( 56,386)
( 57,385)( 58,384)( 59,383)( 60,382)( 61,381)( 62,380)( 63,379)( 64,403)
( 65,405)( 66,404)( 67,400)( 68,402)( 69,401)( 70,397)( 71,399)( 72,398)
( 73,394)( 74,396)( 75,395)( 76,391)( 77,393)( 78,392)( 79,388)( 80,390)
( 81,389)( 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)(109,276)(110,275)(111,274)(112,273)
(113,272)(114,271)(115,279)(116,278)(117,277)(118,292)(119,294)(120,293)
(121,289)(122,291)(123,290)(124,295)(125,297)(126,296)(127,283)(128,285)
(129,284)(130,280)(131,282)(132,281)(133,286)(134,288)(135,287)(136,306)
(137,305)(138,304)(139,303)(140,302)(141,301)(142,300)(143,299)(144,298)
(145,322)(146,324)(147,323)(148,319)(149,321)(150,320)(151,316)(152,318)
(153,317)(154,313)(155,315)(156,314)(157,310)(158,312)(159,311)(160,307)
(161,309)(162,308)(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)(190,445)(191,447)(192,446)
(193,442)(194,444)(195,443)(196,448)(197,450)(198,449)(199,436)(200,438)
(201,437)(202,433)(203,435)(204,434)(205,439)(206,441)(207,440)(208,455)
(209,454)(210,456)(211,452)(212,451)(213,453)(214,458)(215,457)(216,459)
(217,475)(218,477)(219,476)(220,472)(221,474)(222,473)(223,469)(224,471)
(225,470)(226,466)(227,468)(228,467)(229,463)(230,465)(231,464)(232,460)
(233,462)(234,461)(235,485)(236,484)(237,486)(238,482)(239,481)(240,483)
(241,479)(242,478)(243,480);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*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, 32)( 5, 31)( 6, 33)( 7, 36)
( 8, 35)( 9, 34)( 10, 47)( 11, 46)( 12, 48)( 13, 51)( 14, 50)( 15, 49)
( 16, 52)( 17, 54)( 18, 53)( 19, 38)( 20, 37)( 21, 39)( 22, 42)( 23, 41)
( 24, 40)( 25, 43)( 26, 45)( 27, 44)( 56, 57)( 58, 59)( 61, 63)( 64, 74)
( 65, 73)( 66, 75)( 67, 78)( 68, 77)( 69, 76)( 70, 79)( 71, 81)( 72, 80)
( 82,200)( 83,199)( 84,201)( 85,204)( 86,203)( 87,202)( 88,205)( 89,207)
( 90,206)( 91,191)( 92,190)( 93,192)( 94,195)( 95,194)( 96,193)( 97,196)
( 98,198)( 99,197)(100,210)(101,209)(102,208)(103,211)(104,213)(105,212)
(106,215)(107,214)(108,216)(109,173)(110,172)(111,174)(112,177)(113,176)
(114,175)(115,178)(116,180)(117,179)(118,164)(119,163)(120,165)(121,168)
(122,167)(123,166)(124,169)(125,171)(126,170)(127,183)(128,182)(129,181)
(130,184)(131,186)(132,185)(133,188)(134,187)(135,189)(136,227)(137,226)
(138,228)(139,231)(140,230)(141,229)(142,232)(143,234)(144,233)(145,218)
(146,217)(147,219)(148,222)(149,221)(150,220)(151,223)(152,225)(153,224)
(154,237)(155,236)(156,235)(157,238)(158,240)(159,239)(160,242)(161,241)
(162,243)(244,271)(245,273)(246,272)(247,275)(248,274)(249,276)(250,279)
(251,278)(252,277)(253,290)(254,289)(255,291)(256,294)(257,293)(258,292)
(259,295)(260,297)(261,296)(262,281)(263,280)(264,282)(265,285)(266,284)
(267,283)(268,286)(269,288)(270,287)(299,300)(301,302)(304,306)(307,317)
(308,316)(309,318)(310,321)(311,320)(312,319)(313,322)(314,324)(315,323)
(325,443)(326,442)(327,444)(328,447)(329,446)(330,445)(331,448)(332,450)
(333,449)(334,434)(335,433)(336,435)(337,438)(338,437)(339,436)(340,439)
(341,441)(342,440)(343,453)(344,452)(345,451)(346,454)(347,456)(348,455)
(349,458)(350,457)(351,459)(352,416)(353,415)(354,417)(355,420)(356,419)
(357,418)(358,421)(359,423)(360,422)(361,407)(362,406)(363,408)(364,411)
(365,410)(366,409)(367,412)(368,414)(369,413)(370,426)(371,425)(372,424)
(373,427)(374,429)(375,428)(376,431)(377,430)(378,432)(379,470)(380,469)
(381,471)(382,474)(383,473)(384,472)(385,475)(386,477)(387,476)(388,461)
(389,460)(390,462)(391,465)(392,464)(393,463)(394,466)(395,468)(396,467)
(397,480)(398,479)(399,478)(400,481)(401,483)(402,482)(403,485)(404,484)
(405,486);
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)( 28,357)( 29,356)( 30,355)( 31,354)
( 32,353)( 33,352)( 34,360)( 35,359)( 36,358)( 37,373)( 38,375)( 39,374)
( 40,370)( 41,372)( 42,371)( 43,376)( 44,378)( 45,377)( 46,364)( 47,366)
( 48,365)( 49,361)( 50,363)( 51,362)( 52,367)( 53,369)( 54,368)( 55,387)
( 56,386)( 57,385)( 58,384)( 59,383)( 60,382)( 61,381)( 62,380)( 63,379)
( 64,403)( 65,405)( 66,404)( 67,400)( 68,402)( 69,401)( 70,397)( 71,399)
( 72,398)( 73,394)( 74,396)( 75,395)( 76,391)( 77,393)( 78,392)( 79,388)
( 80,390)( 81,389)( 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)(109,276)(110,275)(111,274)
(112,273)(113,272)(114,271)(115,279)(116,278)(117,277)(118,292)(119,294)
(120,293)(121,289)(122,291)(123,290)(124,295)(125,297)(126,296)(127,283)
(128,285)(129,284)(130,280)(131,282)(132,281)(133,286)(134,288)(135,287)
(136,306)(137,305)(138,304)(139,303)(140,302)(141,301)(142,300)(143,299)
(144,298)(145,322)(146,324)(147,323)(148,319)(149,321)(150,320)(151,316)
(152,318)(153,317)(154,313)(155,315)(156,314)(157,310)(158,312)(159,311)
(160,307)(161,309)(162,308)(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)(190,445)(191,447)
(192,446)(193,442)(194,444)(195,443)(196,448)(197,450)(198,449)(199,436)
(200,438)(201,437)(202,433)(203,435)(204,434)(205,439)(206,441)(207,440)
(208,455)(209,454)(210,456)(211,452)(212,451)(213,453)(214,458)(215,457)
(216,459)(217,475)(218,477)(219,476)(220,472)(221,474)(222,473)(223,469)
(224,471)(225,470)(226,466)(227,468)(228,467)(229,463)(230,465)(231,464)
(232,460)(233,462)(234,461)(235,485)(236,484)(237,486)(238,482)(239,481)
(240,483)(241,479)(242,478)(243,480);
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*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope