Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944f
if this polytope has a name.
Group : SmallGroup(1944,943)
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}*972b
3-fold quotients : {18,6}*648c
6-fold quotients : {9,6}*324c
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)( 10,459)( 11,458)( 12,457)( 13,456)( 14,455)( 15,454)( 16,453)
( 17,452)( 18,451)( 19,437)( 20,436)( 21,438)( 22,434)( 23,433)( 24,435)
( 25,440)( 26,439)( 27,441)( 28,428)( 29,427)( 30,429)( 31,425)( 32,424)
( 33,426)( 34,431)( 35,430)( 36,432)( 37,406)( 38,408)( 39,407)( 40,412)
( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,423)( 47,422)( 48,421)
( 49,420)( 50,419)( 51,418)( 52,417)( 53,416)( 54,415)( 55,462)( 56,461)
( 57,460)( 58,468)( 59,467)( 60,466)( 61,465)( 62,464)( 63,463)( 64,476)
( 65,475)( 66,477)( 67,473)( 68,472)( 69,474)( 70,470)( 71,469)( 72,471)
( 73,481)( 74,483)( 75,482)( 76,478)( 77,480)( 78,479)( 79,484)( 80,486)
( 81,485)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)( 88,364)
( 89,366)( 90,365)( 91,378)( 92,377)( 93,376)( 94,375)( 95,374)( 96,373)
( 97,372)( 98,371)( 99,370)(100,356)(101,355)(102,357)(103,353)(104,352)
(105,354)(106,359)(107,358)(108,360)(109,347)(110,346)(111,348)(112,344)
(113,343)(114,345)(115,350)(116,349)(117,351)(118,325)(119,327)(120,326)
(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,342)(128,341)
(129,340)(130,339)(131,338)(132,337)(133,336)(134,335)(135,334)(136,381)
(137,380)(138,379)(139,387)(140,386)(141,385)(142,384)(143,383)(144,382)
(145,395)(146,394)(147,396)(148,392)(149,391)(150,393)(151,389)(152,388)
(153,390)(154,400)(155,402)(156,401)(157,397)(158,399)(159,398)(160,403)
(161,405)(162,404)(163,280)(164,282)(165,281)(166,286)(167,288)(168,287)
(169,283)(170,285)(171,284)(172,297)(173,296)(174,295)(175,294)(176,293)
(177,292)(178,291)(179,290)(180,289)(181,275)(182,274)(183,276)(184,272)
(185,271)(186,273)(187,278)(188,277)(189,279)(190,266)(191,265)(192,267)
(193,263)(194,262)(195,264)(196,269)(197,268)(198,270)(199,244)(200,246)
(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)(208,261)
(209,260)(210,259)(211,258)(212,257)(213,256)(214,255)(215,254)(216,253)
(217,300)(218,299)(219,298)(220,306)(221,305)(222,304)(223,303)(224,302)
(225,301)(226,314)(227,313)(228,315)(229,311)(230,310)(231,312)(232,308)
(233,307)(234,309)(235,319)(236,321)(237,320)(238,316)(239,318)(240,317)
(241,322)(242,324)(243,323);;
s2 := ( 2, 3)( 4, 5)( 7, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)( 14, 22)
( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 28, 56)( 29, 55)( 30, 57)( 31, 60)
( 32, 59)( 33, 58)( 34, 61)( 35, 63)( 36, 62)( 37, 74)( 38, 73)( 39, 75)
( 40, 78)( 41, 77)( 42, 76)( 43, 79)( 44, 81)( 45, 80)( 46, 65)( 47, 64)
( 48, 66)( 49, 69)( 50, 68)( 51, 67)( 52, 70)( 53, 72)( 54, 71)( 83, 84)
( 85, 86)( 88, 90)( 91,100)( 92,102)( 93,101)( 94,104)( 95,103)( 96,105)
( 97,108)( 98,107)( 99,106)(109,137)(110,136)(111,138)(112,141)(113,140)
(114,139)(115,142)(116,144)(117,143)(118,155)(119,154)(120,156)(121,159)
(122,158)(123,157)(124,160)(125,162)(126,161)(127,146)(128,145)(129,147)
(130,150)(131,149)(132,148)(133,151)(134,153)(135,152)(164,165)(166,167)
(169,171)(172,181)(173,183)(174,182)(175,185)(176,184)(177,186)(178,189)
(179,188)(180,187)(190,218)(191,217)(192,219)(193,222)(194,221)(195,220)
(196,223)(197,225)(198,224)(199,236)(200,235)(201,237)(202,240)(203,239)
(204,238)(205,241)(206,243)(207,242)(208,227)(209,226)(210,228)(211,231)
(212,230)(213,229)(214,232)(215,234)(216,233)(245,246)(247,248)(250,252)
(253,262)(254,264)(255,263)(256,266)(257,265)(258,267)(259,270)(260,269)
(261,268)(271,299)(272,298)(273,300)(274,303)(275,302)(276,301)(277,304)
(278,306)(279,305)(280,317)(281,316)(282,318)(283,321)(284,320)(285,319)
(286,322)(287,324)(288,323)(289,308)(290,307)(291,309)(292,312)(293,311)
(294,310)(295,313)(296,315)(297,314)(326,327)(328,329)(331,333)(334,343)
(335,345)(336,344)(337,347)(338,346)(339,348)(340,351)(341,350)(342,349)
(352,380)(353,379)(354,381)(355,384)(356,383)(357,382)(358,385)(359,387)
(360,386)(361,398)(362,397)(363,399)(364,402)(365,401)(366,400)(367,403)
(368,405)(369,404)(370,389)(371,388)(372,390)(373,393)(374,392)(375,391)
(376,394)(377,396)(378,395)(407,408)(409,410)(412,414)(415,424)(416,426)
(417,425)(418,428)(419,427)(420,429)(421,432)(422,431)(423,430)(433,461)
(434,460)(435,462)(436,465)(437,464)(438,463)(439,466)(440,468)(441,467)
(442,479)(443,478)(444,480)(445,483)(446,482)(447,481)(448,484)(449,486)
(450,485)(451,470)(452,469)(453,471)(454,474)(455,473)(456,472)(457,475)
(458,477)(459,476);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*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)( 10,459)( 11,458)( 12,457)( 13,456)( 14,455)( 15,454)
( 16,453)( 17,452)( 18,451)( 19,437)( 20,436)( 21,438)( 22,434)( 23,433)
( 24,435)( 25,440)( 26,439)( 27,441)( 28,428)( 29,427)( 30,429)( 31,425)
( 32,424)( 33,426)( 34,431)( 35,430)( 36,432)( 37,406)( 38,408)( 39,407)
( 40,412)( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,423)( 47,422)
( 48,421)( 49,420)( 50,419)( 51,418)( 52,417)( 53,416)( 54,415)( 55,462)
( 56,461)( 57,460)( 58,468)( 59,467)( 60,466)( 61,465)( 62,464)( 63,463)
( 64,476)( 65,475)( 66,477)( 67,473)( 68,472)( 69,474)( 70,470)( 71,469)
( 72,471)( 73,481)( 74,483)( 75,482)( 76,478)( 77,480)( 78,479)( 79,484)
( 80,486)( 81,485)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)
( 88,364)( 89,366)( 90,365)( 91,378)( 92,377)( 93,376)( 94,375)( 95,374)
( 96,373)( 97,372)( 98,371)( 99,370)(100,356)(101,355)(102,357)(103,353)
(104,352)(105,354)(106,359)(107,358)(108,360)(109,347)(110,346)(111,348)
(112,344)(113,343)(114,345)(115,350)(116,349)(117,351)(118,325)(119,327)
(120,326)(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,342)
(128,341)(129,340)(130,339)(131,338)(132,337)(133,336)(134,335)(135,334)
(136,381)(137,380)(138,379)(139,387)(140,386)(141,385)(142,384)(143,383)
(144,382)(145,395)(146,394)(147,396)(148,392)(149,391)(150,393)(151,389)
(152,388)(153,390)(154,400)(155,402)(156,401)(157,397)(158,399)(159,398)
(160,403)(161,405)(162,404)(163,280)(164,282)(165,281)(166,286)(167,288)
(168,287)(169,283)(170,285)(171,284)(172,297)(173,296)(174,295)(175,294)
(176,293)(177,292)(178,291)(179,290)(180,289)(181,275)(182,274)(183,276)
(184,272)(185,271)(186,273)(187,278)(188,277)(189,279)(190,266)(191,265)
(192,267)(193,263)(194,262)(195,264)(196,269)(197,268)(198,270)(199,244)
(200,246)(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)
(208,261)(209,260)(210,259)(211,258)(212,257)(213,256)(214,255)(215,254)
(216,253)(217,300)(218,299)(219,298)(220,306)(221,305)(222,304)(223,303)
(224,302)(225,301)(226,314)(227,313)(228,315)(229,311)(230,310)(231,312)
(232,308)(233,307)(234,309)(235,319)(236,321)(237,320)(238,316)(239,318)
(240,317)(241,322)(242,324)(243,323);
s2 := Sym(486)!( 2, 3)( 4, 5)( 7, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)
( 14, 22)( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 28, 56)( 29, 55)( 30, 57)
( 31, 60)( 32, 59)( 33, 58)( 34, 61)( 35, 63)( 36, 62)( 37, 74)( 38, 73)
( 39, 75)( 40, 78)( 41, 77)( 42, 76)( 43, 79)( 44, 81)( 45, 80)( 46, 65)
( 47, 64)( 48, 66)( 49, 69)( 50, 68)( 51, 67)( 52, 70)( 53, 72)( 54, 71)
( 83, 84)( 85, 86)( 88, 90)( 91,100)( 92,102)( 93,101)( 94,104)( 95,103)
( 96,105)( 97,108)( 98,107)( 99,106)(109,137)(110,136)(111,138)(112,141)
(113,140)(114,139)(115,142)(116,144)(117,143)(118,155)(119,154)(120,156)
(121,159)(122,158)(123,157)(124,160)(125,162)(126,161)(127,146)(128,145)
(129,147)(130,150)(131,149)(132,148)(133,151)(134,153)(135,152)(164,165)
(166,167)(169,171)(172,181)(173,183)(174,182)(175,185)(176,184)(177,186)
(178,189)(179,188)(180,187)(190,218)(191,217)(192,219)(193,222)(194,221)
(195,220)(196,223)(197,225)(198,224)(199,236)(200,235)(201,237)(202,240)
(203,239)(204,238)(205,241)(206,243)(207,242)(208,227)(209,226)(210,228)
(211,231)(212,230)(213,229)(214,232)(215,234)(216,233)(245,246)(247,248)
(250,252)(253,262)(254,264)(255,263)(256,266)(257,265)(258,267)(259,270)
(260,269)(261,268)(271,299)(272,298)(273,300)(274,303)(275,302)(276,301)
(277,304)(278,306)(279,305)(280,317)(281,316)(282,318)(283,321)(284,320)
(285,319)(286,322)(287,324)(288,323)(289,308)(290,307)(291,309)(292,312)
(293,311)(294,310)(295,313)(296,315)(297,314)(326,327)(328,329)(331,333)
(334,343)(335,345)(336,344)(337,347)(338,346)(339,348)(340,351)(341,350)
(342,349)(352,380)(353,379)(354,381)(355,384)(356,383)(357,382)(358,385)
(359,387)(360,386)(361,398)(362,397)(363,399)(364,402)(365,401)(366,400)
(367,403)(368,405)(369,404)(370,389)(371,388)(372,390)(373,393)(374,392)
(375,391)(376,394)(377,396)(378,395)(407,408)(409,410)(412,414)(415,424)
(416,426)(417,425)(418,428)(419,427)(420,429)(421,432)(422,431)(423,430)
(433,461)(434,460)(435,462)(436,465)(437,464)(438,463)(439,466)(440,468)
(441,467)(442,479)(443,478)(444,480)(445,483)(446,482)(447,481)(448,484)
(449,486)(450,485)(451,470)(452,469)(453,471)(454,474)(455,473)(456,472)
(457,475)(458,477)(459,476);
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 >;
References : None.
to this polytope