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