Polytope of Type {54,6}

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