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Polytope of Type {54,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,18}*1944a
Also Known As : {54,18|2}. if this polytope has another name.
Group : SmallGroup(1944,947)
Rank : 3
Schlafli Type : {54,18}
Number of vertices, edges, etc : 54, 486, 18
Order of s0s1s2 : 54
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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) :
   3-fold quotients : {18,18}*648a, {54,6}*648a
   9-fold quotients : {54,2}*216, {6,18}*216a, {18,6}*216a
   18-fold quotients : {27,2}*108
   27-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a
   54-fold quotients : {2,9}*36, {9,2}*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 := (  2,  3)(  5,  6)(  8,  9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)( 14, 23)
( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)( 37, 48)
( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)
( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)( 68, 77)
( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 82,183)( 83,182)( 84,181)( 85,186)
( 86,185)( 87,184)( 88,189)( 89,188)( 90,187)( 91,174)( 92,173)( 93,172)
( 94,177)( 95,176)( 96,175)( 97,180)( 98,179)( 99,178)(100,165)(101,164)
(102,163)(103,168)(104,167)(105,166)(106,171)(107,170)(108,169)(109,210)
(110,209)(111,208)(112,213)(113,212)(114,211)(115,216)(116,215)(117,214)
(118,201)(119,200)(120,199)(121,204)(122,203)(123,202)(124,207)(125,206)
(126,205)(127,192)(128,191)(129,190)(130,195)(131,194)(132,193)(133,198)
(134,197)(135,196)(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)(245,246)(248,249)(251,252)
(253,264)(254,263)(255,262)(256,267)(257,266)(258,265)(259,270)(260,269)
(261,268)(272,273)(275,276)(278,279)(280,291)(281,290)(282,289)(283,294)
(284,293)(285,292)(286,297)(287,296)(288,295)(299,300)(302,303)(305,306)
(307,318)(308,317)(309,316)(310,321)(311,320)(312,319)(313,324)(314,323)
(315,322)(325,426)(326,425)(327,424)(328,429)(329,428)(330,427)(331,432)
(332,431)(333,430)(334,417)(335,416)(336,415)(337,420)(338,419)(339,418)
(340,423)(341,422)(342,421)(343,408)(344,407)(345,406)(346,411)(347,410)
(348,409)(349,414)(350,413)(351,412)(352,453)(353,452)(354,451)(355,456)
(356,455)(357,454)(358,459)(359,458)(360,457)(361,444)(362,443)(363,442)
(364,447)(365,446)(366,445)(367,450)(368,449)(369,448)(370,435)(371,434)
(372,433)(373,438)(374,437)(375,436)(376,441)(377,440)(378,439)(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);;
s1 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)
(  9, 86)( 10,102)( 11,101)( 12,100)( 13,108)( 14,107)( 15,106)( 16,105)
( 17,104)( 18,103)( 19, 93)( 20, 92)( 21, 91)( 22, 99)( 23, 98)( 24, 97)
( 25, 96)( 26, 95)( 27, 94)( 28,142)( 29,144)( 30,143)( 31,139)( 32,141)
( 33,140)( 34,136)( 35,138)( 36,137)( 37,162)( 38,161)( 39,160)( 40,159)
( 41,158)( 42,157)( 43,156)( 44,155)( 45,154)( 46,153)( 47,152)( 48,151)
( 49,150)( 50,149)( 51,148)( 52,147)( 53,146)( 54,145)( 55,115)( 56,117)
( 57,116)( 58,112)( 59,114)( 60,113)( 61,109)( 62,111)( 63,110)( 64,135)
( 65,134)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,128)( 72,127)
( 73,126)( 74,125)( 75,124)( 76,123)( 77,122)( 78,121)( 79,120)( 80,119)
( 81,118)(163,183)(164,182)(165,181)(166,189)(167,188)(168,187)(169,186)
(170,185)(171,184)(172,174)(175,180)(176,179)(177,178)(190,243)(191,242)
(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,234)
(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)(207,226)
(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)
(216,217)(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)
(251,330)(252,329)(253,345)(254,344)(255,343)(256,351)(257,350)(258,349)
(259,348)(260,347)(261,346)(262,336)(263,335)(264,334)(265,342)(266,341)
(267,340)(268,339)(269,338)(270,337)(271,385)(272,387)(273,386)(274,382)
(275,384)(276,383)(277,379)(278,381)(279,380)(280,405)(281,404)(282,403)
(283,402)(284,401)(285,400)(286,399)(287,398)(288,397)(289,396)(290,395)
(291,394)(292,393)(293,392)(294,391)(295,390)(296,389)(297,388)(298,358)
(299,360)(300,359)(301,355)(302,357)(303,356)(304,352)(305,354)(306,353)
(307,378)(308,377)(309,376)(310,375)(311,374)(312,373)(313,372)(314,371)
(315,370)(316,369)(317,368)(318,367)(319,366)(320,365)(321,364)(322,363)
(323,362)(324,361)(406,426)(407,425)(408,424)(409,432)(410,431)(411,430)
(412,429)(413,428)(414,427)(415,417)(418,423)(419,422)(420,421)(433,486)
(434,485)(435,484)(436,483)(437,482)(438,481)(439,480)(440,479)(441,478)
(442,477)(443,476)(444,475)(445,474)(446,473)(447,472)(448,471)(449,470)
(450,469)(451,468)(452,467)(453,466)(454,465)(455,464)(456,463)(457,462)
(458,461)(459,460);;
s2 := (  1,271)(  2,272)(  3,273)(  4,277)(  5,278)(  6,279)(  7,274)(  8,275)
(  9,276)( 10,280)( 11,281)( 12,282)( 13,286)( 14,287)( 15,288)( 16,283)
( 17,284)( 18,285)( 19,289)( 20,290)( 21,291)( 22,295)( 23,296)( 24,297)
( 25,292)( 26,293)( 27,294)( 28,244)( 29,245)( 30,246)( 31,250)( 32,251)
( 33,252)( 34,247)( 35,248)( 36,249)( 37,253)( 38,254)( 39,255)( 40,259)
( 41,260)( 42,261)( 43,256)( 44,257)( 45,258)( 46,262)( 47,263)( 48,264)
( 49,268)( 50,269)( 51,270)( 52,265)( 53,266)( 54,267)( 55,304)( 56,305)
( 57,306)( 58,301)( 59,302)( 60,303)( 61,298)( 62,299)( 63,300)( 64,313)
( 65,314)( 66,315)( 67,310)( 68,311)( 69,312)( 70,307)( 71,308)( 72,309)
( 73,322)( 74,323)( 75,324)( 76,319)( 77,320)( 78,321)( 79,316)( 80,317)
( 81,318)( 82,352)( 83,353)( 84,354)( 85,358)( 86,359)( 87,360)( 88,355)
( 89,356)( 90,357)( 91,361)( 92,362)( 93,363)( 94,367)( 95,368)( 96,369)
( 97,364)( 98,365)( 99,366)(100,370)(101,371)(102,372)(103,376)(104,377)
(105,378)(106,373)(107,374)(108,375)(109,325)(110,326)(111,327)(112,331)
(113,332)(114,333)(115,328)(116,329)(117,330)(118,334)(119,335)(120,336)
(121,340)(122,341)(123,342)(124,337)(125,338)(126,339)(127,343)(128,344)
(129,345)(130,349)(131,350)(132,351)(133,346)(134,347)(135,348)(136,385)
(137,386)(138,387)(139,382)(140,383)(141,384)(142,379)(143,380)(144,381)
(145,394)(146,395)(147,396)(148,391)(149,392)(150,393)(151,388)(152,389)
(153,390)(154,403)(155,404)(156,405)(157,400)(158,401)(159,402)(160,397)
(161,398)(162,399)(163,433)(164,434)(165,435)(166,439)(167,440)(168,441)
(169,436)(170,437)(171,438)(172,442)(173,443)(174,444)(175,448)(176,449)
(177,450)(178,445)(179,446)(180,447)(181,451)(182,452)(183,453)(184,457)
(185,458)(186,459)(187,454)(188,455)(189,456)(190,406)(191,407)(192,408)
(193,412)(194,413)(195,414)(196,409)(197,410)(198,411)(199,415)(200,416)
(201,417)(202,421)(203,422)(204,423)(205,418)(206,419)(207,420)(208,424)
(209,425)(210,426)(211,430)(212,431)(213,432)(214,427)(215,428)(216,429)
(217,466)(218,467)(219,468)(220,463)(221,464)(222,465)(223,460)(224,461)
(225,462)(226,475)(227,476)(228,477)(229,472)(230,473)(231,474)(232,469)
(233,470)(234,471)(235,484)(236,485)(237,486)(238,481)(239,482)(240,483)
(241,478)(242,479)(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*s2*s1*s0*s1*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(486)!(  2,  3)(  5,  6)(  8,  9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)
( 14, 23)( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)
( 37, 48)( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)
( 45, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)
( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 82,183)( 83,182)( 84,181)
( 85,186)( 86,185)( 87,184)( 88,189)( 89,188)( 90,187)( 91,174)( 92,173)
( 93,172)( 94,177)( 95,176)( 96,175)( 97,180)( 98,179)( 99,178)(100,165)
(101,164)(102,163)(103,168)(104,167)(105,166)(106,171)(107,170)(108,169)
(109,210)(110,209)(111,208)(112,213)(113,212)(114,211)(115,216)(116,215)
(117,214)(118,201)(119,200)(120,199)(121,204)(122,203)(123,202)(124,207)
(125,206)(126,205)(127,192)(128,191)(129,190)(130,195)(131,194)(132,193)
(133,198)(134,197)(135,196)(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)(245,246)(248,249)
(251,252)(253,264)(254,263)(255,262)(256,267)(257,266)(258,265)(259,270)
(260,269)(261,268)(272,273)(275,276)(278,279)(280,291)(281,290)(282,289)
(283,294)(284,293)(285,292)(286,297)(287,296)(288,295)(299,300)(302,303)
(305,306)(307,318)(308,317)(309,316)(310,321)(311,320)(312,319)(313,324)
(314,323)(315,322)(325,426)(326,425)(327,424)(328,429)(329,428)(330,427)
(331,432)(332,431)(333,430)(334,417)(335,416)(336,415)(337,420)(338,419)
(339,418)(340,423)(341,422)(342,421)(343,408)(344,407)(345,406)(346,411)
(347,410)(348,409)(349,414)(350,413)(351,412)(352,453)(353,452)(354,451)
(355,456)(356,455)(357,454)(358,459)(359,458)(360,457)(361,444)(362,443)
(363,442)(364,447)(365,446)(366,445)(367,450)(368,449)(369,448)(370,435)
(371,434)(372,433)(373,438)(374,437)(375,436)(376,441)(377,440)(378,439)
(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);
s1 := Sym(486)!(  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)
(  8, 87)(  9, 86)( 10,102)( 11,101)( 12,100)( 13,108)( 14,107)( 15,106)
( 16,105)( 17,104)( 18,103)( 19, 93)( 20, 92)( 21, 91)( 22, 99)( 23, 98)
( 24, 97)( 25, 96)( 26, 95)( 27, 94)( 28,142)( 29,144)( 30,143)( 31,139)
( 32,141)( 33,140)( 34,136)( 35,138)( 36,137)( 37,162)( 38,161)( 39,160)
( 40,159)( 41,158)( 42,157)( 43,156)( 44,155)( 45,154)( 46,153)( 47,152)
( 48,151)( 49,150)( 50,149)( 51,148)( 52,147)( 53,146)( 54,145)( 55,115)
( 56,117)( 57,116)( 58,112)( 59,114)( 60,113)( 61,109)( 62,111)( 63,110)
( 64,135)( 65,134)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,128)
( 72,127)( 73,126)( 74,125)( 75,124)( 76,123)( 77,122)( 78,121)( 79,120)
( 80,119)( 81,118)(163,183)(164,182)(165,181)(166,189)(167,188)(168,187)
(169,186)(170,185)(171,184)(172,174)(175,180)(176,179)(177,178)(190,243)
(191,242)(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)
(199,234)(200,233)(201,232)(202,231)(203,230)(204,229)(205,228)(206,227)
(207,226)(208,225)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)
(215,218)(216,217)(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)
(250,328)(251,330)(252,329)(253,345)(254,344)(255,343)(256,351)(257,350)
(258,349)(259,348)(260,347)(261,346)(262,336)(263,335)(264,334)(265,342)
(266,341)(267,340)(268,339)(269,338)(270,337)(271,385)(272,387)(273,386)
(274,382)(275,384)(276,383)(277,379)(278,381)(279,380)(280,405)(281,404)
(282,403)(283,402)(284,401)(285,400)(286,399)(287,398)(288,397)(289,396)
(290,395)(291,394)(292,393)(293,392)(294,391)(295,390)(296,389)(297,388)
(298,358)(299,360)(300,359)(301,355)(302,357)(303,356)(304,352)(305,354)
(306,353)(307,378)(308,377)(309,376)(310,375)(311,374)(312,373)(313,372)
(314,371)(315,370)(316,369)(317,368)(318,367)(319,366)(320,365)(321,364)
(322,363)(323,362)(324,361)(406,426)(407,425)(408,424)(409,432)(410,431)
(411,430)(412,429)(413,428)(414,427)(415,417)(418,423)(419,422)(420,421)
(433,486)(434,485)(435,484)(436,483)(437,482)(438,481)(439,480)(440,479)
(441,478)(442,477)(443,476)(444,475)(445,474)(446,473)(447,472)(448,471)
(449,470)(450,469)(451,468)(452,467)(453,466)(454,465)(455,464)(456,463)
(457,462)(458,461)(459,460);
s2 := Sym(486)!(  1,271)(  2,272)(  3,273)(  4,277)(  5,278)(  6,279)(  7,274)
(  8,275)(  9,276)( 10,280)( 11,281)( 12,282)( 13,286)( 14,287)( 15,288)
( 16,283)( 17,284)( 18,285)( 19,289)( 20,290)( 21,291)( 22,295)( 23,296)
( 24,297)( 25,292)( 26,293)( 27,294)( 28,244)( 29,245)( 30,246)( 31,250)
( 32,251)( 33,252)( 34,247)( 35,248)( 36,249)( 37,253)( 38,254)( 39,255)
( 40,259)( 41,260)( 42,261)( 43,256)( 44,257)( 45,258)( 46,262)( 47,263)
( 48,264)( 49,268)( 50,269)( 51,270)( 52,265)( 53,266)( 54,267)( 55,304)
( 56,305)( 57,306)( 58,301)( 59,302)( 60,303)( 61,298)( 62,299)( 63,300)
( 64,313)( 65,314)( 66,315)( 67,310)( 68,311)( 69,312)( 70,307)( 71,308)
( 72,309)( 73,322)( 74,323)( 75,324)( 76,319)( 77,320)( 78,321)( 79,316)
( 80,317)( 81,318)( 82,352)( 83,353)( 84,354)( 85,358)( 86,359)( 87,360)
( 88,355)( 89,356)( 90,357)( 91,361)( 92,362)( 93,363)( 94,367)( 95,368)
( 96,369)( 97,364)( 98,365)( 99,366)(100,370)(101,371)(102,372)(103,376)
(104,377)(105,378)(106,373)(107,374)(108,375)(109,325)(110,326)(111,327)
(112,331)(113,332)(114,333)(115,328)(116,329)(117,330)(118,334)(119,335)
(120,336)(121,340)(122,341)(123,342)(124,337)(125,338)(126,339)(127,343)
(128,344)(129,345)(130,349)(131,350)(132,351)(133,346)(134,347)(135,348)
(136,385)(137,386)(138,387)(139,382)(140,383)(141,384)(142,379)(143,380)
(144,381)(145,394)(146,395)(147,396)(148,391)(149,392)(150,393)(151,388)
(152,389)(153,390)(154,403)(155,404)(156,405)(157,400)(158,401)(159,402)
(160,397)(161,398)(162,399)(163,433)(164,434)(165,435)(166,439)(167,440)
(168,441)(169,436)(170,437)(171,438)(172,442)(173,443)(174,444)(175,448)
(176,449)(177,450)(178,445)(179,446)(180,447)(181,451)(182,452)(183,453)
(184,457)(185,458)(186,459)(187,454)(188,455)(189,456)(190,406)(191,407)
(192,408)(193,412)(194,413)(195,414)(196,409)(197,410)(198,411)(199,415)
(200,416)(201,417)(202,421)(203,422)(204,423)(205,418)(206,419)(207,420)
(208,424)(209,425)(210,426)(211,430)(212,431)(213,432)(214,427)(215,428)
(216,429)(217,466)(218,467)(219,468)(220,463)(221,464)(222,465)(223,460)
(224,461)(225,462)(226,475)(227,476)(228,477)(229,472)(230,473)(231,474)
(232,469)(233,470)(234,471)(235,484)(236,485)(237,486)(238,481)(239,482)
(240,483)(241,478)(242,479)(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*s2*s1*s0*s1*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope