Polytope of Type {6,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*1944a
if this polytope has a name.
Group : SmallGroup(1944,948)
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}*972a
3-fold quotients : {6,18}*648a, {6,54}*648b
6-fold quotients : {6,9}*324a, {6,27}*324
9-fold quotients : {2,54}*216, {6,18}*216b, {6,6}*216a
18-fold quotients : {2,27}*108, {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, 31)( 5, 33)( 6, 32)( 7, 34)( 8, 36)
( 9, 35)( 10, 48)( 11, 47)( 12, 46)( 13, 51)( 14, 50)( 15, 49)( 16, 54)
( 17, 53)( 18, 52)( 19, 39)( 20, 38)( 21, 37)( 22, 42)( 23, 41)( 24, 40)
( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)
( 66, 73)( 67, 78)( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 82,210)
( 83,209)( 84,208)( 85,213)( 86,212)( 87,211)( 88,216)( 89,215)( 90,214)
( 91,201)( 92,200)( 93,199)( 94,204)( 95,203)( 96,202)( 97,207)( 98,206)
( 99,205)(100,192)(101,191)(102,190)(103,195)(104,194)(105,193)(106,198)
(107,197)(108,196)(109,183)(110,182)(111,181)(112,186)(113,185)(114,184)
(115,189)(116,188)(117,187)(118,174)(119,173)(120,172)(121,177)(122,176)
(123,175)(124,180)(125,179)(126,178)(127,165)(128,164)(129,163)(130,168)
(131,167)(132,166)(133,171)(134,170)(135,169)(136,237)(137,236)(138,235)
(139,240)(140,239)(141,238)(142,243)(143,242)(144,241)(145,228)(146,227)
(147,226)(148,231)(149,230)(150,229)(151,234)(152,233)(153,232)(154,219)
(155,218)(156,217)(157,222)(158,221)(159,220)(160,225)(161,224)(162,223)
(244,271)(245,273)(246,272)(247,274)(248,276)(249,275)(250,277)(251,279)
(252,278)(253,291)(254,290)(255,289)(256,294)(257,293)(258,292)(259,297)
(260,296)(261,295)(262,282)(263,281)(264,280)(265,285)(266,284)(267,283)
(268,288)(269,287)(270,286)(299,300)(302,303)(305,306)(307,318)(308,317)
(309,316)(310,321)(311,320)(312,319)(313,324)(314,323)(315,322)(325,453)
(326,452)(327,451)(328,456)(329,455)(330,454)(331,459)(332,458)(333,457)
(334,444)(335,443)(336,442)(337,447)(338,446)(339,445)(340,450)(341,449)
(342,448)(343,435)(344,434)(345,433)(346,438)(347,437)(348,436)(349,441)
(350,440)(351,439)(352,426)(353,425)(354,424)(355,429)(356,428)(357,427)
(358,432)(359,431)(360,430)(361,417)(362,416)(363,415)(364,420)(365,419)
(366,418)(367,423)(368,422)(369,421)(370,408)(371,407)(372,406)(373,411)
(374,410)(375,409)(376,414)(377,413)(378,412)(379,480)(380,479)(381,478)
(382,483)(383,482)(384,481)(385,486)(386,485)(387,484)(388,471)(389,470)
(390,469)(391,474)(392,473)(393,472)(394,477)(395,476)(396,475)(397,462)
(398,461)(399,460)(400,465)(401,464)(402,463)(403,468)(404,467)(405,466);;
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)( 28,385)( 29,387)( 30,386)( 31,379)( 32,381)
( 33,380)( 34,382)( 35,384)( 36,383)( 37,405)( 38,404)( 39,403)( 40,399)
( 41,398)( 42,397)( 43,402)( 44,401)( 45,400)( 46,396)( 47,395)( 48,394)
( 49,390)( 50,389)( 51,388)( 52,393)( 53,392)( 54,391)( 55,355)( 56,357)
( 57,356)( 58,358)( 59,360)( 60,359)( 61,352)( 62,354)( 63,353)( 64,375)
( 65,374)( 66,373)( 67,378)( 68,377)( 69,376)( 70,372)( 71,371)( 72,370)
( 73,366)( 74,365)( 75,364)( 76,369)( 77,368)( 78,367)( 79,363)( 80,362)
( 81,361)( 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)(109,304)(110,306)(111,305)(112,298)
(113,300)(114,299)(115,301)(116,303)(117,302)(118,324)(119,323)(120,322)
(121,318)(122,317)(123,316)(124,321)(125,320)(126,319)(127,315)(128,314)
(129,313)(130,309)(131,308)(132,307)(133,312)(134,311)(135,310)(136,274)
(137,276)(138,275)(139,277)(140,279)(141,278)(142,271)(143,273)(144,272)
(145,294)(146,293)(147,292)(148,297)(149,296)(150,295)(151,291)(152,290)
(153,289)(154,285)(155,284)(156,283)(157,288)(158,287)(159,286)(160,282)
(161,281)(162,280)(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)(190,486)(191,485)(192,484)
(193,480)(194,479)(195,478)(196,483)(197,482)(198,481)(199,477)(200,476)
(201,475)(202,471)(203,470)(204,469)(205,474)(206,473)(207,472)(208,468)
(209,467)(210,466)(211,462)(212,461)(213,460)(214,465)(215,464)(216,463)
(217,456)(218,455)(219,454)(220,459)(221,458)(222,457)(223,453)(224,452)
(225,451)(226,447)(227,446)(228,445)(229,450)(230,449)(231,448)(232,444)
(233,443)(234,442)(235,438)(236,437)(237,436)(238,441)(239,440)(240,439)
(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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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, 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, 31)( 5, 33)( 6, 32)( 7, 34)
( 8, 36)( 9, 35)( 10, 48)( 11, 47)( 12, 46)( 13, 51)( 14, 50)( 15, 49)
( 16, 54)( 17, 53)( 18, 52)( 19, 39)( 20, 38)( 21, 37)( 22, 42)( 23, 41)
( 24, 40)( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 59, 60)( 62, 63)( 64, 75)
( 65, 74)( 66, 73)( 67, 78)( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)
( 82,210)( 83,209)( 84,208)( 85,213)( 86,212)( 87,211)( 88,216)( 89,215)
( 90,214)( 91,201)( 92,200)( 93,199)( 94,204)( 95,203)( 96,202)( 97,207)
( 98,206)( 99,205)(100,192)(101,191)(102,190)(103,195)(104,194)(105,193)
(106,198)(107,197)(108,196)(109,183)(110,182)(111,181)(112,186)(113,185)
(114,184)(115,189)(116,188)(117,187)(118,174)(119,173)(120,172)(121,177)
(122,176)(123,175)(124,180)(125,179)(126,178)(127,165)(128,164)(129,163)
(130,168)(131,167)(132,166)(133,171)(134,170)(135,169)(136,237)(137,236)
(138,235)(139,240)(140,239)(141,238)(142,243)(143,242)(144,241)(145,228)
(146,227)(147,226)(148,231)(149,230)(150,229)(151,234)(152,233)(153,232)
(154,219)(155,218)(156,217)(157,222)(158,221)(159,220)(160,225)(161,224)
(162,223)(244,271)(245,273)(246,272)(247,274)(248,276)(249,275)(250,277)
(251,279)(252,278)(253,291)(254,290)(255,289)(256,294)(257,293)(258,292)
(259,297)(260,296)(261,295)(262,282)(263,281)(264,280)(265,285)(266,284)
(267,283)(268,288)(269,287)(270,286)(299,300)(302,303)(305,306)(307,318)
(308,317)(309,316)(310,321)(311,320)(312,319)(313,324)(314,323)(315,322)
(325,453)(326,452)(327,451)(328,456)(329,455)(330,454)(331,459)(332,458)
(333,457)(334,444)(335,443)(336,442)(337,447)(338,446)(339,445)(340,450)
(341,449)(342,448)(343,435)(344,434)(345,433)(346,438)(347,437)(348,436)
(349,441)(350,440)(351,439)(352,426)(353,425)(354,424)(355,429)(356,428)
(357,427)(358,432)(359,431)(360,430)(361,417)(362,416)(363,415)(364,420)
(365,419)(366,418)(367,423)(368,422)(369,421)(370,408)(371,407)(372,406)
(373,411)(374,410)(375,409)(376,414)(377,413)(378,412)(379,480)(380,479)
(381,478)(382,483)(383,482)(384,481)(385,486)(386,485)(387,484)(388,471)
(389,470)(390,469)(391,474)(392,473)(393,472)(394,477)(395,476)(396,475)
(397,462)(398,461)(399,460)(400,465)(401,464)(402,463)(403,468)(404,467)
(405,466);
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)( 28,385)( 29,387)( 30,386)( 31,379)
( 32,381)( 33,380)( 34,382)( 35,384)( 36,383)( 37,405)( 38,404)( 39,403)
( 40,399)( 41,398)( 42,397)( 43,402)( 44,401)( 45,400)( 46,396)( 47,395)
( 48,394)( 49,390)( 50,389)( 51,388)( 52,393)( 53,392)( 54,391)( 55,355)
( 56,357)( 57,356)( 58,358)( 59,360)( 60,359)( 61,352)( 62,354)( 63,353)
( 64,375)( 65,374)( 66,373)( 67,378)( 68,377)( 69,376)( 70,372)( 71,371)
( 72,370)( 73,366)( 74,365)( 75,364)( 76,369)( 77,368)( 78,367)( 79,363)
( 80,362)( 81,361)( 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)(109,304)(110,306)(111,305)
(112,298)(113,300)(114,299)(115,301)(116,303)(117,302)(118,324)(119,323)
(120,322)(121,318)(122,317)(123,316)(124,321)(125,320)(126,319)(127,315)
(128,314)(129,313)(130,309)(131,308)(132,307)(133,312)(134,311)(135,310)
(136,274)(137,276)(138,275)(139,277)(140,279)(141,278)(142,271)(143,273)
(144,272)(145,294)(146,293)(147,292)(148,297)(149,296)(150,295)(151,291)
(152,290)(153,289)(154,285)(155,284)(156,283)(157,288)(158,287)(159,286)
(160,282)(161,281)(162,280)(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)(190,486)(191,485)
(192,484)(193,480)(194,479)(195,478)(196,483)(197,482)(198,481)(199,477)
(200,476)(201,475)(202,471)(203,470)(204,469)(205,474)(206,473)(207,472)
(208,468)(209,467)(210,466)(211,462)(212,461)(213,460)(214,465)(215,464)
(216,463)(217,456)(218,455)(219,454)(220,459)(221,458)(222,457)(223,453)
(224,452)(225,451)(226,447)(227,446)(228,445)(229,450)(230,449)(231,448)
(232,444)(233,443)(234,442)(235,438)(236,437)(237,436)(238,441)(239,440)
(240,439)(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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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