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Polytope of Type {6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1944g
if this polytope has a name.
Group : SmallGroup(1944,949)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 54, 486, 162
Order of s0s1s2 : 18
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 : {6,18}*972c
   3-fold quotients : {6,18}*648b, {6,6}*648b
   6-fold quotients : {6,18}*324a, {6,6}*324a
   9-fold quotients : {6,18}*216a, {6,6}*216b
   18-fold quotients : {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 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)
(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)
(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)
(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)(277,301)
(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)(285,314)
(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)(293,324)
(294,323)(295,319)(296,321)(297,320)(326,327)(328,331)(329,333)(330,332)
(335,336)(337,340)(338,342)(339,341)(344,345)(346,349)(347,351)(348,350)
(352,379)(353,381)(354,380)(355,385)(356,387)(357,386)(358,382)(359,384)
(360,383)(361,388)(362,390)(363,389)(364,394)(365,396)(366,395)(367,391)
(368,393)(369,392)(370,397)(371,399)(372,398)(373,403)(374,405)(375,404)
(376,400)(377,402)(378,401)(407,408)(409,412)(410,414)(411,413)(416,417)
(418,421)(419,423)(420,422)(425,426)(427,430)(428,432)(429,431)(433,460)
(434,462)(435,461)(436,466)(437,468)(438,467)(439,463)(440,465)(441,464)
(442,469)(443,471)(444,470)(445,475)(446,477)(447,476)(448,472)(449,474)
(450,473)(451,478)(452,480)(453,479)(454,484)(455,486)(456,485)(457,481)
(458,483)(459,482);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)(  8, 36)
(  9, 35)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 51)( 15, 50)( 16, 52)
( 17, 54)( 18, 53)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)( 24, 41)
( 25, 43)( 26, 45)( 27, 44)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)
( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 82,199)
( 83,201)( 84,200)( 85,202)( 86,204)( 87,203)( 88,205)( 89,207)( 90,206)
( 91,190)( 92,192)( 93,191)( 94,193)( 95,195)( 96,194)( 97,196)( 98,198)
( 99,197)(100,208)(101,210)(102,209)(103,211)(104,213)(105,212)(106,214)
(107,216)(108,215)(109,172)(110,174)(111,173)(112,175)(113,177)(114,176)
(115,178)(116,180)(117,179)(118,163)(119,165)(120,164)(121,166)(122,168)
(123,167)(124,169)(125,171)(126,170)(127,181)(128,183)(129,182)(130,184)
(131,186)(132,185)(133,187)(134,189)(135,188)(136,226)(137,228)(138,227)
(139,229)(140,231)(141,230)(142,232)(143,234)(144,233)(145,217)(146,219)
(147,218)(148,220)(149,222)(150,221)(151,223)(152,225)(153,224)(154,235)
(155,237)(156,236)(157,238)(158,240)(159,239)(160,241)(161,243)(162,242)
(244,271)(245,273)(246,272)(247,274)(248,276)(249,275)(250,277)(251,279)
(252,278)(253,289)(254,291)(255,290)(256,292)(257,294)(258,293)(259,295)
(260,297)(261,296)(262,280)(263,282)(264,281)(265,283)(266,285)(267,284)
(268,286)(269,288)(270,287)(299,300)(302,303)(305,306)(307,316)(308,318)
(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)(315,323)(325,442)
(326,444)(327,443)(328,445)(329,447)(330,446)(331,448)(332,450)(333,449)
(334,433)(335,435)(336,434)(337,436)(338,438)(339,437)(340,439)(341,441)
(342,440)(343,451)(344,453)(345,452)(346,454)(347,456)(348,455)(349,457)
(350,459)(351,458)(352,415)(353,417)(354,416)(355,418)(356,420)(357,419)
(358,421)(359,423)(360,422)(361,406)(362,408)(363,407)(364,409)(365,411)
(366,410)(367,412)(368,414)(369,413)(370,424)(371,426)(372,425)(373,427)
(374,429)(375,428)(376,430)(377,432)(378,431)(379,469)(380,471)(381,470)
(382,472)(383,474)(384,473)(385,475)(386,477)(387,476)(388,460)(389,462)
(390,461)(391,463)(392,465)(393,464)(394,466)(395,468)(396,467)(397,478)
(398,480)(399,479)(400,481)(401,483)(402,482)(403,484)(404,486)(405,485);;
s2 := (  1,325)(  2,326)(  3,327)(  4,332)(  5,333)(  6,331)(  7,330)(  8,328)
(  9,329)( 10,343)( 11,344)( 12,345)( 13,350)( 14,351)( 15,349)( 16,348)
( 17,346)( 18,347)( 19,334)( 20,335)( 21,336)( 22,341)( 23,342)( 24,340)
( 25,339)( 26,337)( 27,338)( 28,356)( 29,357)( 30,355)( 31,354)( 32,352)
( 33,353)( 34,358)( 35,359)( 36,360)( 37,374)( 38,375)( 39,373)( 40,372)
( 41,370)( 42,371)( 43,376)( 44,377)( 45,378)( 46,365)( 47,366)( 48,364)
( 49,363)( 50,361)( 51,362)( 52,367)( 53,368)( 54,369)( 55,387)( 56,385)
( 57,386)( 58,382)( 59,383)( 60,384)( 61,380)( 62,381)( 63,379)( 64,405)
( 65,403)( 66,404)( 67,400)( 68,401)( 69,402)( 70,398)( 71,399)( 72,397)
( 73,396)( 74,394)( 75,395)( 76,391)( 77,392)( 78,393)( 79,389)( 80,390)
( 81,388)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)( 88,249)
( 89,247)( 90,248)( 91,262)( 92,263)( 93,264)( 94,269)( 95,270)( 96,268)
( 97,267)( 98,265)( 99,266)(100,253)(101,254)(102,255)(103,260)(104,261)
(105,259)(106,258)(107,256)(108,257)(109,275)(110,276)(111,274)(112,273)
(113,271)(114,272)(115,277)(116,278)(117,279)(118,293)(119,294)(120,292)
(121,291)(122,289)(123,290)(124,295)(125,296)(126,297)(127,284)(128,285)
(129,283)(130,282)(131,280)(132,281)(133,286)(134,287)(135,288)(136,306)
(137,304)(138,305)(139,301)(140,302)(141,303)(142,299)(143,300)(144,298)
(145,324)(146,322)(147,323)(148,319)(149,320)(150,321)(151,317)(152,318)
(153,316)(154,315)(155,313)(156,314)(157,310)(158,311)(159,312)(160,308)
(161,309)(162,307)(163,415)(164,416)(165,417)(166,422)(167,423)(168,421)
(169,420)(170,418)(171,419)(172,406)(173,407)(174,408)(175,413)(176,414)
(177,412)(178,411)(179,409)(180,410)(181,424)(182,425)(183,426)(184,431)
(185,432)(186,430)(187,429)(188,427)(189,428)(190,446)(191,447)(192,445)
(193,444)(194,442)(195,443)(196,448)(197,449)(198,450)(199,437)(200,438)
(201,436)(202,435)(203,433)(204,434)(205,439)(206,440)(207,441)(208,455)
(209,456)(210,454)(211,453)(212,451)(213,452)(214,457)(215,458)(216,459)
(217,477)(218,475)(219,476)(220,472)(221,473)(222,474)(223,470)(224,471)
(225,469)(226,468)(227,466)(228,467)(229,463)(230,464)(231,465)(232,461)
(233,462)(234,460)(235,486)(236,484)(237,485)(238,481)(239,482)(240,483)
(241,479)(242,480)(243,478);;
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(486)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)
(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)
(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)
(211,241)(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)
(277,301)(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)
(285,314)(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)
(293,324)(294,323)(295,319)(296,321)(297,320)(326,327)(328,331)(329,333)
(330,332)(335,336)(337,340)(338,342)(339,341)(344,345)(346,349)(347,351)
(348,350)(352,379)(353,381)(354,380)(355,385)(356,387)(357,386)(358,382)
(359,384)(360,383)(361,388)(362,390)(363,389)(364,394)(365,396)(366,395)
(367,391)(368,393)(369,392)(370,397)(371,399)(372,398)(373,403)(374,405)
(375,404)(376,400)(377,402)(378,401)(407,408)(409,412)(410,414)(411,413)
(416,417)(418,421)(419,423)(420,422)(425,426)(427,430)(428,432)(429,431)
(433,460)(434,462)(435,461)(436,466)(437,468)(438,467)(439,463)(440,465)
(441,464)(442,469)(443,471)(444,470)(445,475)(446,477)(447,476)(448,472)
(449,474)(450,473)(451,478)(452,480)(453,479)(454,484)(455,486)(456,485)
(457,481)(458,483)(459,482);
s1 := Sym(486)!(  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)
(  8, 36)(  9, 35)( 10, 46)( 11, 48)( 12, 47)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 54)( 18, 53)( 19, 37)( 20, 39)( 21, 38)( 22, 40)( 23, 42)
( 24, 41)( 25, 43)( 26, 45)( 27, 44)( 56, 57)( 59, 60)( 62, 63)( 64, 73)
( 65, 75)( 66, 74)( 67, 76)( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)
( 82,199)( 83,201)( 84,200)( 85,202)( 86,204)( 87,203)( 88,205)( 89,207)
( 90,206)( 91,190)( 92,192)( 93,191)( 94,193)( 95,195)( 96,194)( 97,196)
( 98,198)( 99,197)(100,208)(101,210)(102,209)(103,211)(104,213)(105,212)
(106,214)(107,216)(108,215)(109,172)(110,174)(111,173)(112,175)(113,177)
(114,176)(115,178)(116,180)(117,179)(118,163)(119,165)(120,164)(121,166)
(122,168)(123,167)(124,169)(125,171)(126,170)(127,181)(128,183)(129,182)
(130,184)(131,186)(132,185)(133,187)(134,189)(135,188)(136,226)(137,228)
(138,227)(139,229)(140,231)(141,230)(142,232)(143,234)(144,233)(145,217)
(146,219)(147,218)(148,220)(149,222)(150,221)(151,223)(152,225)(153,224)
(154,235)(155,237)(156,236)(157,238)(158,240)(159,239)(160,241)(161,243)
(162,242)(244,271)(245,273)(246,272)(247,274)(248,276)(249,275)(250,277)
(251,279)(252,278)(253,289)(254,291)(255,290)(256,292)(257,294)(258,293)
(259,295)(260,297)(261,296)(262,280)(263,282)(264,281)(265,283)(266,285)
(267,284)(268,286)(269,288)(270,287)(299,300)(302,303)(305,306)(307,316)
(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)(315,323)
(325,442)(326,444)(327,443)(328,445)(329,447)(330,446)(331,448)(332,450)
(333,449)(334,433)(335,435)(336,434)(337,436)(338,438)(339,437)(340,439)
(341,441)(342,440)(343,451)(344,453)(345,452)(346,454)(347,456)(348,455)
(349,457)(350,459)(351,458)(352,415)(353,417)(354,416)(355,418)(356,420)
(357,419)(358,421)(359,423)(360,422)(361,406)(362,408)(363,407)(364,409)
(365,411)(366,410)(367,412)(368,414)(369,413)(370,424)(371,426)(372,425)
(373,427)(374,429)(375,428)(376,430)(377,432)(378,431)(379,469)(380,471)
(381,470)(382,472)(383,474)(384,473)(385,475)(386,477)(387,476)(388,460)
(389,462)(390,461)(391,463)(392,465)(393,464)(394,466)(395,468)(396,467)
(397,478)(398,480)(399,479)(400,481)(401,483)(402,482)(403,484)(404,486)
(405,485);
s2 := Sym(486)!(  1,325)(  2,326)(  3,327)(  4,332)(  5,333)(  6,331)(  7,330)
(  8,328)(  9,329)( 10,343)( 11,344)( 12,345)( 13,350)( 14,351)( 15,349)
( 16,348)( 17,346)( 18,347)( 19,334)( 20,335)( 21,336)( 22,341)( 23,342)
( 24,340)( 25,339)( 26,337)( 27,338)( 28,356)( 29,357)( 30,355)( 31,354)
( 32,352)( 33,353)( 34,358)( 35,359)( 36,360)( 37,374)( 38,375)( 39,373)
( 40,372)( 41,370)( 42,371)( 43,376)( 44,377)( 45,378)( 46,365)( 47,366)
( 48,364)( 49,363)( 50,361)( 51,362)( 52,367)( 53,368)( 54,369)( 55,387)
( 56,385)( 57,386)( 58,382)( 59,383)( 60,384)( 61,380)( 62,381)( 63,379)
( 64,405)( 65,403)( 66,404)( 67,400)( 68,401)( 69,402)( 70,398)( 71,399)
( 72,397)( 73,396)( 74,394)( 75,395)( 76,391)( 77,392)( 78,393)( 79,389)
( 80,390)( 81,388)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)
( 88,249)( 89,247)( 90,248)( 91,262)( 92,263)( 93,264)( 94,269)( 95,270)
( 96,268)( 97,267)( 98,265)( 99,266)(100,253)(101,254)(102,255)(103,260)
(104,261)(105,259)(106,258)(107,256)(108,257)(109,275)(110,276)(111,274)
(112,273)(113,271)(114,272)(115,277)(116,278)(117,279)(118,293)(119,294)
(120,292)(121,291)(122,289)(123,290)(124,295)(125,296)(126,297)(127,284)
(128,285)(129,283)(130,282)(131,280)(132,281)(133,286)(134,287)(135,288)
(136,306)(137,304)(138,305)(139,301)(140,302)(141,303)(142,299)(143,300)
(144,298)(145,324)(146,322)(147,323)(148,319)(149,320)(150,321)(151,317)
(152,318)(153,316)(154,315)(155,313)(156,314)(157,310)(158,311)(159,312)
(160,308)(161,309)(162,307)(163,415)(164,416)(165,417)(166,422)(167,423)
(168,421)(169,420)(170,418)(171,419)(172,406)(173,407)(174,408)(175,413)
(176,414)(177,412)(178,411)(179,409)(180,410)(181,424)(182,425)(183,426)
(184,431)(185,432)(186,430)(187,429)(188,427)(189,428)(190,446)(191,447)
(192,445)(193,444)(194,442)(195,443)(196,448)(197,449)(198,450)(199,437)
(200,438)(201,436)(202,435)(203,433)(204,434)(205,439)(206,440)(207,441)
(208,455)(209,456)(210,454)(211,453)(212,451)(213,452)(214,457)(215,458)
(216,459)(217,477)(218,475)(219,476)(220,472)(221,473)(222,474)(223,470)
(224,471)(225,469)(226,468)(227,466)(228,467)(229,463)(230,464)(231,465)
(232,461)(233,462)(234,460)(235,486)(236,484)(237,485)(238,481)(239,482)
(240,483)(241,479)(242,480)(243,478);
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*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 >; 
 
References : None.
to this polytope