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