Polytope of Type {6,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*1944b
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 : {6,54}*972a
3-fold quotients : {6,18}*648b, {6,54}*648a
6-fold quotients : {6,18}*324a
9-fold quotients : {2,54}*216, {6,18}*216a, {6,6}*216b
18-fold quotients : {2,27}*108, {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, 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)( 4,331)( 5,333)( 6,332)( 7,328)( 8,330)
( 9,329)( 10,345)( 11,344)( 12,343)( 13,351)( 14,350)( 15,349)( 16,348)
( 17,347)( 18,346)( 19,336)( 20,335)( 21,334)( 22,342)( 23,341)( 24,340)
( 25,339)( 26,338)( 27,337)( 28,355)( 29,357)( 30,356)( 31,352)( 32,354)
( 33,353)( 34,358)( 35,360)( 36,359)( 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)( 55,385)( 56,387)
( 57,386)( 58,382)( 59,384)( 60,383)( 61,379)( 62,381)( 63,380)( 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)( 91,264)( 92,263)( 93,262)( 94,270)( 95,269)( 96,268)
( 97,267)( 98,266)( 99,265)(100,255)(101,254)(102,253)(103,261)(104,260)
(105,259)(106,258)(107,257)(108,256)(109,274)(110,276)(111,275)(112,271)
(113,273)(114,272)(115,277)(116,279)(117,278)(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)(136,304)
(137,306)(138,305)(139,301)(140,303)(141,302)(142,298)(143,300)(144,299)
(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)(163,426)(164,425)(165,424)(166,432)(167,431)(168,430)
(169,429)(170,428)(171,427)(172,417)(173,416)(174,415)(175,423)(176,422)
(177,421)(178,420)(179,419)(180,418)(181,408)(182,407)(183,406)(184,414)
(185,413)(186,412)(187,411)(188,410)(189,409)(190,456)(191,455)(192,454)
(193,453)(194,452)(195,451)(196,459)(197,458)(198,457)(199,447)(200,446)
(201,445)(202,444)(203,443)(204,442)(205,450)(206,449)(207,448)(208,438)
(209,437)(210,436)(211,435)(212,434)(213,433)(214,441)(215,440)(216,439)
(217,486)(218,485)(219,484)(220,483)(221,482)(222,481)(223,480)(224,479)
(225,478)(226,477)(227,476)(228,475)(229,474)(230,473)(231,472)(232,471)
(233,470)(234,469)(235,468)(236,467)(237,466)(238,465)(239,464)(240,463)
(241,462)(242,461)(243,460);;
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*s2*s1*s0*s1*s2*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, 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)( 4,331)( 5,333)( 6,332)( 7,328)
( 8,330)( 9,329)( 10,345)( 11,344)( 12,343)( 13,351)( 14,350)( 15,349)
( 16,348)( 17,347)( 18,346)( 19,336)( 20,335)( 21,334)( 22,342)( 23,341)
( 24,340)( 25,339)( 26,338)( 27,337)( 28,355)( 29,357)( 30,356)( 31,352)
( 32,354)( 33,353)( 34,358)( 35,360)( 36,359)( 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)( 55,385)
( 56,387)( 57,386)( 58,382)( 59,384)( 60,383)( 61,379)( 62,381)( 63,380)
( 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)( 91,264)( 92,263)( 93,262)( 94,270)( 95,269)
( 96,268)( 97,267)( 98,266)( 99,265)(100,255)(101,254)(102,253)(103,261)
(104,260)(105,259)(106,258)(107,257)(108,256)(109,274)(110,276)(111,275)
(112,271)(113,273)(114,272)(115,277)(116,279)(117,278)(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)
(136,304)(137,306)(138,305)(139,301)(140,303)(141,302)(142,298)(143,300)
(144,299)(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)(163,426)(164,425)(165,424)(166,432)(167,431)
(168,430)(169,429)(170,428)(171,427)(172,417)(173,416)(174,415)(175,423)
(176,422)(177,421)(178,420)(179,419)(180,418)(181,408)(182,407)(183,406)
(184,414)(185,413)(186,412)(187,411)(188,410)(189,409)(190,456)(191,455)
(192,454)(193,453)(194,452)(195,451)(196,459)(197,458)(198,457)(199,447)
(200,446)(201,445)(202,444)(203,443)(204,442)(205,450)(206,449)(207,448)
(208,438)(209,437)(210,436)(211,435)(212,434)(213,433)(214,441)(215,440)
(216,439)(217,486)(218,485)(219,484)(220,483)(221,482)(222,481)(223,480)
(224,479)(225,478)(226,477)(227,476)(228,475)(229,474)(230,473)(231,472)
(232,471)(233,470)(234,469)(235,468)(236,467)(237,466)(238,465)(239,464)
(240,463)(241,462)(242,461)(243,460);
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*s2*s1*s0*s1*s2*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