Polytope of Type {54,18}

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