Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944h
if this polytope has a name.
Group : SmallGroup(1944,944)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 6
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}*972c
3-fold quotients : {18,6}*648c, {6,18}*648h
6-fold quotients : {9,6}*324c, {3,18}*324
9-fold quotients : {6,6}*216c
18-fold quotients : {3,6}*108
27-fold quotients : {6,6}*72c
54-fold quotients : {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)( 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)( 82,163)
( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)( 89,168)( 90,167)
( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)( 98,177)
( 99,176)(100,181)(101,183)(102,182)(103,187)(104,189)(105,188)(106,184)
(107,186)(108,185)(109,218)(110,217)(111,219)(112,224)(113,223)(114,225)
(115,221)(116,220)(117,222)(118,227)(119,226)(120,228)(121,233)(122,232)
(123,234)(124,230)(125,229)(126,231)(127,236)(128,235)(129,237)(130,242)
(131,241)(132,243)(133,239)(134,238)(135,240)(136,191)(137,190)(138,192)
(139,197)(140,196)(141,198)(142,194)(143,193)(144,195)(145,200)(146,199)
(147,201)(148,206)(149,205)(150,207)(151,203)(152,202)(153,204)(154,209)
(155,208)(156,210)(157,215)(158,214)(159,216)(160,212)(161,211)(162,213)
(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)(325,406)
(326,408)(327,407)(328,412)(329,414)(330,413)(331,409)(332,411)(333,410)
(334,415)(335,417)(336,416)(337,421)(338,423)(339,422)(340,418)(341,420)
(342,419)(343,424)(344,426)(345,425)(346,430)(347,432)(348,431)(349,427)
(350,429)(351,428)(352,461)(353,460)(354,462)(355,467)(356,466)(357,468)
(358,464)(359,463)(360,465)(361,470)(362,469)(363,471)(364,476)(365,475)
(366,477)(367,473)(368,472)(369,474)(370,479)(371,478)(372,480)(373,485)
(374,484)(375,486)(376,482)(377,481)(378,483)(379,434)(380,433)(381,435)
(382,440)(383,439)(384,441)(385,437)(386,436)(387,438)(388,443)(389,442)
(390,444)(391,449)(392,448)(393,450)(394,446)(395,445)(396,447)(397,452)
(398,451)(399,453)(400,458)(401,457)(402,459)(403,455)(404,454)(405,456);;
s1 := ( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)( 8,447)
( 9,446)( 10,458)( 11,457)( 12,459)( 13,455)( 14,454)( 15,456)( 16,452)
( 17,451)( 18,453)( 19,438)( 20,437)( 21,436)( 22,435)( 23,434)( 24,433)
( 25,441)( 26,440)( 27,439)( 28,429)( 29,428)( 30,427)( 31,426)( 32,425)
( 33,424)( 34,432)( 35,431)( 36,430)( 37,406)( 38,408)( 39,407)( 40,412)
( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,422)( 47,421)( 48,423)
( 49,419)( 50,418)( 51,420)( 52,416)( 53,415)( 54,417)( 55,460)( 56,462)
( 57,461)( 58,466)( 59,468)( 60,467)( 61,463)( 62,465)( 63,464)( 64,476)
( 65,475)( 66,477)( 67,473)( 68,472)( 69,474)( 70,470)( 71,469)( 72,471)
( 73,483)( 74,482)( 75,481)( 76,480)( 77,479)( 78,478)( 79,486)( 80,485)
( 81,484)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)( 88,364)
( 89,366)( 90,365)( 91,377)( 92,376)( 93,378)( 94,374)( 95,373)( 96,375)
( 97,371)( 98,370)( 99,372)(100,357)(101,356)(102,355)(103,354)(104,353)
(105,352)(106,360)(107,359)(108,358)(109,348)(110,347)(111,346)(112,345)
(113,344)(114,343)(115,351)(116,350)(117,349)(118,325)(119,327)(120,326)
(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,341)(128,340)
(129,342)(130,338)(131,337)(132,339)(133,335)(134,334)(135,336)(136,379)
(137,381)(138,380)(139,385)(140,387)(141,386)(142,382)(143,384)(144,383)
(145,395)(146,394)(147,396)(148,392)(149,391)(150,393)(151,389)(152,388)
(153,390)(154,402)(155,401)(156,400)(157,399)(158,398)(159,397)(160,405)
(161,404)(162,403)(163,280)(164,282)(165,281)(166,286)(167,288)(168,287)
(169,283)(170,285)(171,284)(172,296)(173,295)(174,297)(175,293)(176,292)
(177,294)(178,290)(179,289)(180,291)(181,276)(182,275)(183,274)(184,273)
(185,272)(186,271)(187,279)(188,278)(189,277)(190,267)(191,266)(192,265)
(193,264)(194,263)(195,262)(196,270)(197,269)(198,268)(199,244)(200,246)
(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)(208,260)
(209,259)(210,261)(211,257)(212,256)(213,258)(214,254)(215,253)(216,255)
(217,298)(218,300)(219,299)(220,304)(221,306)(222,305)(223,301)(224,303)
(225,302)(226,314)(227,313)(228,315)(229,311)(230,310)(231,312)(232,308)
(233,307)(234,309)(235,321)(236,320)(237,319)(238,318)(239,317)(240,316)
(241,324)(242,323)(243,322);;
s2 := ( 2, 3)( 4, 6)( 7, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 56)( 29, 55)( 30, 57)( 31, 58)
( 32, 60)( 33, 59)( 34, 63)( 35, 62)( 36, 61)( 37, 74)( 38, 73)( 39, 75)
( 40, 76)( 41, 78)( 42, 77)( 43, 81)( 44, 80)( 45, 79)( 46, 65)( 47, 64)
( 48, 66)( 49, 67)( 50, 69)( 51, 68)( 52, 72)( 53, 71)( 54, 70)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,137)(110,136)(111,138)(112,139)(113,141)
(114,140)(115,144)(116,143)(117,142)(118,155)(119,154)(120,156)(121,157)
(122,159)(123,158)(124,162)(125,161)(126,160)(127,146)(128,145)(129,147)
(130,148)(131,150)(132,149)(133,153)(134,152)(135,151)(164,165)(166,168)
(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)
(179,187)(180,189)(190,218)(191,217)(192,219)(193,220)(194,222)(195,221)
(196,225)(197,224)(198,223)(199,236)(200,235)(201,237)(202,238)(203,240)
(204,239)(205,243)(206,242)(207,241)(208,227)(209,226)(210,228)(211,229)
(212,231)(213,230)(214,234)(215,233)(216,232)(245,246)(247,249)(250,251)
(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)
(261,270)(271,299)(272,298)(273,300)(274,301)(275,303)(276,302)(277,306)
(278,305)(279,304)(280,317)(281,316)(282,318)(283,319)(284,321)(285,320)
(286,324)(287,323)(288,322)(289,308)(290,307)(291,309)(292,310)(293,312)
(294,311)(295,315)(296,314)(297,313)(326,327)(328,330)(331,332)(334,343)
(335,345)(336,344)(337,348)(338,347)(339,346)(340,350)(341,349)(342,351)
(352,380)(353,379)(354,381)(355,382)(356,384)(357,383)(358,387)(359,386)
(360,385)(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)(367,405)
(368,404)(369,403)(370,389)(371,388)(372,390)(373,391)(374,393)(375,392)
(376,396)(377,395)(378,394)(407,408)(409,411)(412,413)(415,424)(416,426)
(417,425)(418,429)(419,428)(420,427)(421,431)(422,430)(423,432)(433,461)
(434,460)(435,462)(436,463)(437,465)(438,464)(439,468)(440,467)(441,466)
(442,479)(443,478)(444,480)(445,481)(446,483)(447,482)(448,486)(449,485)
(450,484)(451,470)(452,469)(453,471)(454,472)(455,474)(456,473)(457,477)
(458,476)(459,475);;
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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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 ];;
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, 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)
( 82,163)( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)( 89,168)
( 90,167)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)
( 98,177)( 99,176)(100,181)(101,183)(102,182)(103,187)(104,189)(105,188)
(106,184)(107,186)(108,185)(109,218)(110,217)(111,219)(112,224)(113,223)
(114,225)(115,221)(116,220)(117,222)(118,227)(119,226)(120,228)(121,233)
(122,232)(123,234)(124,230)(125,229)(126,231)(127,236)(128,235)(129,237)
(130,242)(131,241)(132,243)(133,239)(134,238)(135,240)(136,191)(137,190)
(138,192)(139,197)(140,196)(141,198)(142,194)(143,193)(144,195)(145,200)
(146,199)(147,201)(148,206)(149,205)(150,207)(151,203)(152,202)(153,204)
(154,209)(155,208)(156,210)(157,215)(158,214)(159,216)(160,212)(161,211)
(162,213)(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)
(325,406)(326,408)(327,407)(328,412)(329,414)(330,413)(331,409)(332,411)
(333,410)(334,415)(335,417)(336,416)(337,421)(338,423)(339,422)(340,418)
(341,420)(342,419)(343,424)(344,426)(345,425)(346,430)(347,432)(348,431)
(349,427)(350,429)(351,428)(352,461)(353,460)(354,462)(355,467)(356,466)
(357,468)(358,464)(359,463)(360,465)(361,470)(362,469)(363,471)(364,476)
(365,475)(366,477)(367,473)(368,472)(369,474)(370,479)(371,478)(372,480)
(373,485)(374,484)(375,486)(376,482)(377,481)(378,483)(379,434)(380,433)
(381,435)(382,440)(383,439)(384,441)(385,437)(386,436)(387,438)(388,443)
(389,442)(390,444)(391,449)(392,448)(393,450)(394,446)(395,445)(396,447)
(397,452)(398,451)(399,453)(400,458)(401,457)(402,459)(403,455)(404,454)
(405,456);
s1 := Sym(486)!( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)
( 8,447)( 9,446)( 10,458)( 11,457)( 12,459)( 13,455)( 14,454)( 15,456)
( 16,452)( 17,451)( 18,453)( 19,438)( 20,437)( 21,436)( 22,435)( 23,434)
( 24,433)( 25,441)( 26,440)( 27,439)( 28,429)( 29,428)( 30,427)( 31,426)
( 32,425)( 33,424)( 34,432)( 35,431)( 36,430)( 37,406)( 38,408)( 39,407)
( 40,412)( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,422)( 47,421)
( 48,423)( 49,419)( 50,418)( 51,420)( 52,416)( 53,415)( 54,417)( 55,460)
( 56,462)( 57,461)( 58,466)( 59,468)( 60,467)( 61,463)( 62,465)( 63,464)
( 64,476)( 65,475)( 66,477)( 67,473)( 68,472)( 69,474)( 70,470)( 71,469)
( 72,471)( 73,483)( 74,482)( 75,481)( 76,480)( 77,479)( 78,478)( 79,486)
( 80,485)( 81,484)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)
( 88,364)( 89,366)( 90,365)( 91,377)( 92,376)( 93,378)( 94,374)( 95,373)
( 96,375)( 97,371)( 98,370)( 99,372)(100,357)(101,356)(102,355)(103,354)
(104,353)(105,352)(106,360)(107,359)(108,358)(109,348)(110,347)(111,346)
(112,345)(113,344)(114,343)(115,351)(116,350)(117,349)(118,325)(119,327)
(120,326)(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,341)
(128,340)(129,342)(130,338)(131,337)(132,339)(133,335)(134,334)(135,336)
(136,379)(137,381)(138,380)(139,385)(140,387)(141,386)(142,382)(143,384)
(144,383)(145,395)(146,394)(147,396)(148,392)(149,391)(150,393)(151,389)
(152,388)(153,390)(154,402)(155,401)(156,400)(157,399)(158,398)(159,397)
(160,405)(161,404)(162,403)(163,280)(164,282)(165,281)(166,286)(167,288)
(168,287)(169,283)(170,285)(171,284)(172,296)(173,295)(174,297)(175,293)
(176,292)(177,294)(178,290)(179,289)(180,291)(181,276)(182,275)(183,274)
(184,273)(185,272)(186,271)(187,279)(188,278)(189,277)(190,267)(191,266)
(192,265)(193,264)(194,263)(195,262)(196,270)(197,269)(198,268)(199,244)
(200,246)(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)
(208,260)(209,259)(210,261)(211,257)(212,256)(213,258)(214,254)(215,253)
(216,255)(217,298)(218,300)(219,299)(220,304)(221,306)(222,305)(223,301)
(224,303)(225,302)(226,314)(227,313)(228,315)(229,311)(230,310)(231,312)
(232,308)(233,307)(234,309)(235,321)(236,320)(237,319)(238,318)(239,317)
(240,316)(241,324)(242,323)(243,322);
s2 := Sym(486)!( 2, 3)( 4, 6)( 7, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)
( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 56)( 29, 55)( 30, 57)
( 31, 58)( 32, 60)( 33, 59)( 34, 63)( 35, 62)( 36, 61)( 37, 74)( 38, 73)
( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 81)( 44, 80)( 45, 79)( 46, 65)
( 47, 64)( 48, 66)( 49, 67)( 50, 69)( 51, 68)( 52, 72)( 53, 71)( 54, 70)
( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)
( 96,103)( 97,107)( 98,106)( 99,108)(109,137)(110,136)(111,138)(112,139)
(113,141)(114,140)(115,144)(116,143)(117,142)(118,155)(119,154)(120,156)
(121,157)(122,159)(123,158)(124,162)(125,161)(126,160)(127,146)(128,145)
(129,147)(130,148)(131,150)(132,149)(133,153)(134,152)(135,151)(164,165)
(166,168)(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)
(178,188)(179,187)(180,189)(190,218)(191,217)(192,219)(193,220)(194,222)
(195,221)(196,225)(197,224)(198,223)(199,236)(200,235)(201,237)(202,238)
(203,240)(204,239)(205,243)(206,242)(207,241)(208,227)(209,226)(210,228)
(211,229)(212,231)(213,230)(214,234)(215,233)(216,232)(245,246)(247,249)
(250,251)(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)
(260,268)(261,270)(271,299)(272,298)(273,300)(274,301)(275,303)(276,302)
(277,306)(278,305)(279,304)(280,317)(281,316)(282,318)(283,319)(284,321)
(285,320)(286,324)(287,323)(288,322)(289,308)(290,307)(291,309)(292,310)
(293,312)(294,311)(295,315)(296,314)(297,313)(326,327)(328,330)(331,332)
(334,343)(335,345)(336,344)(337,348)(338,347)(339,346)(340,350)(341,349)
(342,351)(352,380)(353,379)(354,381)(355,382)(356,384)(357,383)(358,387)
(359,386)(360,385)(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)
(367,405)(368,404)(369,403)(370,389)(371,388)(372,390)(373,391)(374,393)
(375,392)(376,396)(377,395)(378,394)(407,408)(409,411)(412,413)(415,424)
(416,426)(417,425)(418,429)(419,428)(420,427)(421,431)(422,430)(423,432)
(433,461)(434,460)(435,462)(436,463)(437,465)(438,464)(439,468)(440,467)
(441,466)(442,479)(443,478)(444,480)(445,481)(446,483)(447,482)(448,486)
(449,485)(450,484)(451,470)(452,469)(453,471)(454,472)(455,474)(456,473)
(457,477)(458,476)(459,475);
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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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 >;
References : None.
to this polytope