Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944y
if this polytope has a name.
Group : SmallGroup(1944,951)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
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 : {9,18}*972i
   3-fold quotients : {18,6}*648a, {6,18}*648h
   6-fold quotients : {9,6}*324a, {3,18}*324
   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, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 28, 56)( 29, 55)( 30, 57)( 31, 59)
( 32, 58)( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 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)( 82,172)
( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)( 89,180)( 90,179)
( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167)( 97,169)( 98,171)
( 99,170)(100,181)(101,183)(102,182)(103,184)(104,186)(105,185)(106,187)
(107,189)(108,188)(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)(127,236)(128,235)(129,237)(130,239)
(131,238)(132,240)(133,242)(134,241)(135,243)(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)(154,209)
(155,208)(156,210)(157,212)(158,211)(159,213)(160,215)(161,214)(162,216)
(245,246)(248,249)(251,252)(253,262)(254,264)(255,263)(256,265)(257,267)
(258,266)(259,268)(260,270)(261,269)(271,299)(272,298)(273,300)(274,302)
(275,301)(276,303)(277,305)(278,304)(279,306)(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)(325,415)
(326,417)(327,416)(328,418)(329,420)(330,419)(331,421)(332,423)(333,422)
(334,406)(335,408)(336,407)(337,409)(338,411)(339,410)(340,412)(341,414)
(342,413)(343,424)(344,426)(345,425)(346,427)(347,429)(348,428)(349,430)
(350,432)(351,431)(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)(370,479)(371,478)(372,480)(373,482)
(374,481)(375,483)(376,485)(377,484)(378,486)(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)(397,452)
(398,451)(399,453)(400,455)(401,454)(402,456)(403,458)(404,457)(405,459);;
s1 := (  1,436)(  2,438)(  3,437)(  4,441)(  5,440)(  6,439)(  7,434)(  8,433)
(  9,435)( 10,454)( 11,456)( 12,455)( 13,459)( 14,458)( 15,457)( 16,452)
( 17,451)( 18,453)( 19,445)( 20,447)( 21,446)( 22,450)( 23,449)( 24,448)
( 25,443)( 26,442)( 27,444)( 28,413)( 29,412)( 30,414)( 31,406)( 32,408)
( 33,407)( 34,411)( 35,410)( 36,409)( 37,431)( 38,430)( 39,432)( 40,424)
( 41,426)( 42,425)( 43,429)( 44,428)( 45,427)( 46,422)( 47,421)( 48,423)
( 49,415)( 50,417)( 51,416)( 52,420)( 53,419)( 54,418)( 55,460)( 56,462)
( 57,461)( 58,465)( 59,464)( 60,463)( 61,467)( 62,466)( 63,468)( 64,478)
( 65,480)( 66,479)( 67,483)( 68,482)( 69,481)( 70,485)( 71,484)( 72,486)
( 73,469)( 74,471)( 75,470)( 76,474)( 77,473)( 78,472)( 79,476)( 80,475)
( 81,477)( 82,355)( 83,357)( 84,356)( 85,360)( 86,359)( 87,358)( 88,353)
( 89,352)( 90,354)( 91,373)( 92,375)( 93,374)( 94,378)( 95,377)( 96,376)
( 97,371)( 98,370)( 99,372)(100,364)(101,366)(102,365)(103,369)(104,368)
(105,367)(106,362)(107,361)(108,363)(109,332)(110,331)(111,333)(112,325)
(113,327)(114,326)(115,330)(116,329)(117,328)(118,350)(119,349)(120,351)
(121,343)(122,345)(123,344)(124,348)(125,347)(126,346)(127,341)(128,340)
(129,342)(130,334)(131,336)(132,335)(133,339)(134,338)(135,337)(136,379)
(137,381)(138,380)(139,384)(140,383)(141,382)(142,386)(143,385)(144,387)
(145,397)(146,399)(147,398)(148,402)(149,401)(150,400)(151,404)(152,403)
(153,405)(154,388)(155,390)(156,389)(157,393)(158,392)(159,391)(160,395)
(161,394)(162,396)(163,274)(164,276)(165,275)(166,279)(167,278)(168,277)
(169,272)(170,271)(171,273)(172,292)(173,294)(174,293)(175,297)(176,296)
(177,295)(178,290)(179,289)(180,291)(181,283)(182,285)(183,284)(184,288)
(185,287)(186,286)(187,281)(188,280)(189,282)(190,251)(191,250)(192,252)
(193,244)(194,246)(195,245)(196,249)(197,248)(198,247)(199,269)(200,268)
(201,270)(202,262)(203,264)(204,263)(205,267)(206,266)(207,265)(208,260)
(209,259)(210,261)(211,253)(212,255)(213,254)(214,258)(215,257)(216,256)
(217,298)(218,300)(219,299)(220,303)(221,302)(222,301)(223,305)(224,304)
(225,306)(226,316)(227,318)(228,317)(229,321)(230,320)(231,319)(232,323)
(233,322)(234,324)(235,307)(236,309)(237,308)(238,312)(239,311)(240,310)
(241,314)(242,313)(243,315);;
s2 := (  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, 56)( 29, 55)( 30, 57)( 31, 62)
( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)( 39, 66)
( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)( 47, 73)
( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,137)(110,136)(111,138)(112,143)(113,142)
(114,144)(115,140)(116,139)(117,141)(118,146)(119,145)(120,147)(121,152)
(122,151)(123,153)(124,149)(125,148)(126,150)(127,155)(128,154)(129,156)
(130,161)(131,160)(132,162)(133,158)(134,157)(135,159)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,218)(191,217)(192,219)(193,224)(194,223)(195,225)
(196,221)(197,220)(198,222)(199,227)(200,226)(201,228)(202,233)(203,232)
(204,234)(205,230)(206,229)(207,231)(208,236)(209,235)(210,237)(211,242)
(212,241)(213,243)(214,239)(215,238)(216,240)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,299)(272,298)(273,300)(274,305)(275,304)(276,306)(277,302)
(278,301)(279,303)(280,308)(281,307)(282,309)(283,314)(284,313)(285,315)
(286,311)(287,310)(288,312)(289,317)(290,316)(291,318)(292,323)(293,322)
(294,324)(295,320)(296,319)(297,321)(326,327)(328,331)(329,333)(330,332)
(335,336)(337,340)(338,342)(339,341)(344,345)(346,349)(347,351)(348,350)
(352,380)(353,379)(354,381)(355,386)(356,385)(357,387)(358,383)(359,382)
(360,384)(361,389)(362,388)(363,390)(364,395)(365,394)(366,396)(367,392)
(368,391)(369,393)(370,398)(371,397)(372,399)(373,404)(374,403)(375,405)
(376,401)(377,400)(378,402)(407,408)(409,412)(410,414)(411,413)(416,417)
(418,421)(419,423)(420,422)(425,426)(427,430)(428,432)(429,431)(433,461)
(434,460)(435,462)(436,467)(437,466)(438,468)(439,464)(440,463)(441,465)
(442,470)(443,469)(444,471)(445,476)(446,475)(447,477)(448,473)(449,472)
(450,474)(451,479)(452,478)(453,480)(454,485)(455,484)(456,486)(457,482)
(458,481)(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, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(486)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 28, 56)( 29, 55)( 30, 57)
( 31, 59)( 32, 58)( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 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)
( 82,172)( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)( 89,180)
( 90,179)( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167)( 97,169)
( 98,171)( 99,170)(100,181)(101,183)(102,182)(103,184)(104,186)(105,185)
(106,187)(107,189)(108,188)(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)(127,236)(128,235)(129,237)
(130,239)(131,238)(132,240)(133,242)(134,241)(135,243)(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)
(154,209)(155,208)(156,210)(157,212)(158,211)(159,213)(160,215)(161,214)
(162,216)(245,246)(248,249)(251,252)(253,262)(254,264)(255,263)(256,265)
(257,267)(258,266)(259,268)(260,270)(261,269)(271,299)(272,298)(273,300)
(274,302)(275,301)(276,303)(277,305)(278,304)(279,306)(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)
(325,415)(326,417)(327,416)(328,418)(329,420)(330,419)(331,421)(332,423)
(333,422)(334,406)(335,408)(336,407)(337,409)(338,411)(339,410)(340,412)
(341,414)(342,413)(343,424)(344,426)(345,425)(346,427)(347,429)(348,428)
(349,430)(350,432)(351,431)(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)(370,479)(371,478)(372,480)
(373,482)(374,481)(375,483)(376,485)(377,484)(378,486)(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)
(397,452)(398,451)(399,453)(400,455)(401,454)(402,456)(403,458)(404,457)
(405,459);
s1 := Sym(486)!(  1,436)(  2,438)(  3,437)(  4,441)(  5,440)(  6,439)(  7,434)
(  8,433)(  9,435)( 10,454)( 11,456)( 12,455)( 13,459)( 14,458)( 15,457)
( 16,452)( 17,451)( 18,453)( 19,445)( 20,447)( 21,446)( 22,450)( 23,449)
( 24,448)( 25,443)( 26,442)( 27,444)( 28,413)( 29,412)( 30,414)( 31,406)
( 32,408)( 33,407)( 34,411)( 35,410)( 36,409)( 37,431)( 38,430)( 39,432)
( 40,424)( 41,426)( 42,425)( 43,429)( 44,428)( 45,427)( 46,422)( 47,421)
( 48,423)( 49,415)( 50,417)( 51,416)( 52,420)( 53,419)( 54,418)( 55,460)
( 56,462)( 57,461)( 58,465)( 59,464)( 60,463)( 61,467)( 62,466)( 63,468)
( 64,478)( 65,480)( 66,479)( 67,483)( 68,482)( 69,481)( 70,485)( 71,484)
( 72,486)( 73,469)( 74,471)( 75,470)( 76,474)( 77,473)( 78,472)( 79,476)
( 80,475)( 81,477)( 82,355)( 83,357)( 84,356)( 85,360)( 86,359)( 87,358)
( 88,353)( 89,352)( 90,354)( 91,373)( 92,375)( 93,374)( 94,378)( 95,377)
( 96,376)( 97,371)( 98,370)( 99,372)(100,364)(101,366)(102,365)(103,369)
(104,368)(105,367)(106,362)(107,361)(108,363)(109,332)(110,331)(111,333)
(112,325)(113,327)(114,326)(115,330)(116,329)(117,328)(118,350)(119,349)
(120,351)(121,343)(122,345)(123,344)(124,348)(125,347)(126,346)(127,341)
(128,340)(129,342)(130,334)(131,336)(132,335)(133,339)(134,338)(135,337)
(136,379)(137,381)(138,380)(139,384)(140,383)(141,382)(142,386)(143,385)
(144,387)(145,397)(146,399)(147,398)(148,402)(149,401)(150,400)(151,404)
(152,403)(153,405)(154,388)(155,390)(156,389)(157,393)(158,392)(159,391)
(160,395)(161,394)(162,396)(163,274)(164,276)(165,275)(166,279)(167,278)
(168,277)(169,272)(170,271)(171,273)(172,292)(173,294)(174,293)(175,297)
(176,296)(177,295)(178,290)(179,289)(180,291)(181,283)(182,285)(183,284)
(184,288)(185,287)(186,286)(187,281)(188,280)(189,282)(190,251)(191,250)
(192,252)(193,244)(194,246)(195,245)(196,249)(197,248)(198,247)(199,269)
(200,268)(201,270)(202,262)(203,264)(204,263)(205,267)(206,266)(207,265)
(208,260)(209,259)(210,261)(211,253)(212,255)(213,254)(214,258)(215,257)
(216,256)(217,298)(218,300)(219,299)(220,303)(221,302)(222,301)(223,305)
(224,304)(225,306)(226,316)(227,318)(228,317)(229,321)(230,320)(231,319)
(232,323)(233,322)(234,324)(235,307)(236,309)(237,308)(238,312)(239,311)
(240,310)(241,314)(242,313)(243,315);
s2 := 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, 56)( 29, 55)( 30, 57)
( 31, 62)( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)
( 39, 66)( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)
( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,137)(110,136)(111,138)(112,143)
(113,142)(114,144)(115,140)(116,139)(117,141)(118,146)(119,145)(120,147)
(121,152)(122,151)(123,153)(124,149)(125,148)(126,150)(127,155)(128,154)
(129,156)(130,161)(131,160)(132,162)(133,158)(134,157)(135,159)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,218)(191,217)(192,219)(193,224)(194,223)
(195,225)(196,221)(197,220)(198,222)(199,227)(200,226)(201,228)(202,233)
(203,232)(204,234)(205,230)(206,229)(207,231)(208,236)(209,235)(210,237)
(211,242)(212,241)(213,243)(214,239)(215,238)(216,240)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,299)(272,298)(273,300)(274,305)(275,304)(276,306)
(277,302)(278,301)(279,303)(280,308)(281,307)(282,309)(283,314)(284,313)
(285,315)(286,311)(287,310)(288,312)(289,317)(290,316)(291,318)(292,323)
(293,322)(294,324)(295,320)(296,319)(297,321)(326,327)(328,331)(329,333)
(330,332)(335,336)(337,340)(338,342)(339,341)(344,345)(346,349)(347,351)
(348,350)(352,380)(353,379)(354,381)(355,386)(356,385)(357,387)(358,383)
(359,382)(360,384)(361,389)(362,388)(363,390)(364,395)(365,394)(366,396)
(367,392)(368,391)(369,393)(370,398)(371,397)(372,399)(373,404)(374,403)
(375,405)(376,401)(377,400)(378,402)(407,408)(409,412)(410,414)(411,413)
(416,417)(418,421)(419,423)(420,422)(425,426)(427,430)(428,432)(429,431)
(433,461)(434,460)(435,462)(436,467)(437,466)(438,468)(439,464)(440,463)
(441,465)(442,470)(443,469)(444,471)(445,476)(446,475)(447,477)(448,473)
(449,472)(450,474)(451,479)(452,478)(453,480)(454,485)(455,484)(456,486)
(457,482)(458,481)(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, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*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 >;
References : None.
to this polytope