Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944k
if this polytope has a name.
Group : SmallGroup(1944,945)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
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 : {18,18}*972c
3-fold quotients : {18,6}*648f, {6,18}*648g
6-fold quotients : {18,6}*324b, {6,18}*324c
9-fold quotients : {6,6}*216b
18-fold quotients : {6,6}*108
27-fold quotients : {6,6}*72a
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, 6)( 7, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 57)( 29, 56)( 30, 55)( 31, 59)
( 32, 58)( 33, 60)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)( 39, 73)
( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)( 47, 65)
( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,138)(110,137)(111,136)(112,140)(113,139)
(114,141)(115,142)(116,144)(117,143)(118,156)(119,155)(120,154)(121,158)
(122,157)(123,159)(124,160)(125,162)(126,161)(127,147)(128,146)(129,145)
(130,149)(131,148)(132,150)(133,151)(134,153)(135,152)(164,165)(166,168)
(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)
(179,187)(180,189)(190,219)(191,218)(192,217)(193,221)(194,220)(195,222)
(196,223)(197,225)(198,224)(199,237)(200,236)(201,235)(202,239)(203,238)
(204,240)(205,241)(206,243)(207,242)(208,228)(209,227)(210,226)(211,230)
(212,229)(213,231)(214,232)(215,234)(216,233)(245,246)(247,249)(250,251)
(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)
(261,270)(271,300)(272,299)(273,298)(274,302)(275,301)(276,303)(277,304)
(278,306)(279,305)(280,318)(281,317)(282,316)(283,320)(284,319)(285,321)
(286,322)(287,324)(288,323)(289,309)(290,308)(291,307)(292,311)(293,310)
(294,312)(295,313)(296,315)(297,314)(326,327)(328,330)(331,332)(334,343)
(335,345)(336,344)(337,348)(338,347)(339,346)(340,350)(341,349)(342,351)
(352,381)(353,380)(354,379)(355,383)(356,382)(357,384)(358,385)(359,387)
(360,386)(361,399)(362,398)(363,397)(364,401)(365,400)(366,402)(367,403)
(368,405)(369,404)(370,390)(371,389)(372,388)(373,392)(374,391)(375,393)
(376,394)(377,396)(378,395)(407,408)(409,411)(412,413)(415,424)(416,426)
(417,425)(418,429)(419,428)(420,427)(421,431)(422,430)(423,432)(433,462)
(434,461)(435,460)(436,464)(437,463)(438,465)(439,466)(440,468)(441,467)
(442,480)(443,479)(444,478)(445,482)(446,481)(447,483)(448,484)(449,486)
(450,485)(451,471)(452,470)(453,469)(454,473)(455,472)(456,474)(457,475)
(458,477)(459,476);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)( 8, 33)
( 9, 32)( 10, 45)( 11, 44)( 12, 43)( 13, 42)( 14, 41)( 15, 40)( 16, 39)
( 17, 38)( 18, 37)( 19, 50)( 20, 49)( 21, 51)( 22, 47)( 23, 46)( 24, 48)
( 25, 53)( 26, 52)( 27, 54)( 55, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 71)
( 65, 70)( 66, 72)( 67, 68)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82,194)
( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)( 88,197)( 89,196)( 90,198)
( 91,199)( 92,201)( 93,200)( 94,205)( 95,207)( 96,206)( 97,202)( 98,204)
( 99,203)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)(106,210)
(107,209)(108,208)(109,167)(110,166)(111,168)(112,164)(113,163)(114,165)
(115,170)(116,169)(117,171)(118,172)(119,174)(120,173)(121,178)(122,180)
(123,179)(124,175)(125,177)(126,176)(127,189)(128,188)(129,187)(130,186)
(131,185)(132,184)(133,183)(134,182)(135,181)(136,220)(137,222)(138,221)
(139,217)(140,219)(141,218)(142,223)(143,225)(144,224)(145,228)(146,227)
(147,226)(148,234)(149,233)(150,232)(151,231)(152,230)(153,229)(154,242)
(155,241)(156,243)(157,239)(158,238)(159,240)(160,236)(161,235)(162,237)
(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)(250,274)(251,276)
(252,275)(253,288)(254,287)(255,286)(256,285)(257,284)(258,283)(259,282)
(260,281)(261,280)(262,293)(263,292)(264,294)(265,290)(266,289)(267,291)
(268,296)(269,295)(270,297)(298,300)(301,306)(302,305)(303,304)(307,314)
(308,313)(309,315)(310,311)(316,319)(317,321)(318,320)(323,324)(325,437)
(326,436)(327,438)(328,434)(329,433)(330,435)(331,440)(332,439)(333,441)
(334,442)(335,444)(336,443)(337,448)(338,450)(339,449)(340,445)(341,447)
(342,446)(343,459)(344,458)(345,457)(346,456)(347,455)(348,454)(349,453)
(350,452)(351,451)(352,410)(353,409)(354,411)(355,407)(356,406)(357,408)
(358,413)(359,412)(360,414)(361,415)(362,417)(363,416)(364,421)(365,423)
(366,422)(367,418)(368,420)(369,419)(370,432)(371,431)(372,430)(373,429)
(374,428)(375,427)(376,426)(377,425)(378,424)(379,463)(380,465)(381,464)
(382,460)(383,462)(384,461)(385,466)(386,468)(387,467)(388,471)(389,470)
(390,469)(391,477)(392,476)(393,475)(394,474)(395,473)(396,472)(397,485)
(398,484)(399,486)(400,482)(401,481)(402,483)(403,479)(404,478)(405,480);;
s2 := ( 1,325)( 2,326)( 3,327)( 4,332)( 5,333)( 6,331)( 7,330)( 8,328)
( 9,329)( 10,345)( 11,343)( 12,344)( 13,349)( 14,350)( 15,351)( 16,347)
( 17,348)( 18,346)( 19,335)( 20,336)( 21,334)( 22,342)( 23,340)( 24,341)
( 25,337)( 26,338)( 27,339)( 28,364)( 29,365)( 30,366)( 31,362)( 32,363)
( 33,361)( 34,369)( 35,367)( 36,368)( 37,357)( 38,355)( 39,356)( 40,352)
( 41,353)( 42,354)( 43,359)( 44,360)( 45,358)( 46,374)( 47,375)( 48,373)
( 49,372)( 50,370)( 51,371)( 52,376)( 53,377)( 54,378)( 55,402)( 56,400)
( 57,401)( 58,397)( 59,398)( 60,399)( 61,404)( 62,405)( 63,403)( 64,392)
( 65,393)( 66,391)( 67,390)( 68,388)( 69,389)( 70,394)( 71,395)( 72,396)
( 73,382)( 74,383)( 75,384)( 76,380)( 77,381)( 78,379)( 79,387)( 80,385)
( 81,386)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)( 88,249)
( 89,247)( 90,248)( 91,264)( 92,262)( 93,263)( 94,268)( 95,269)( 96,270)
( 97,266)( 98,267)( 99,265)(100,254)(101,255)(102,253)(103,261)(104,259)
(105,260)(106,256)(107,257)(108,258)(109,283)(110,284)(111,285)(112,281)
(113,282)(114,280)(115,288)(116,286)(117,287)(118,276)(119,274)(120,275)
(121,271)(122,272)(123,273)(124,278)(125,279)(126,277)(127,293)(128,294)
(129,292)(130,291)(131,289)(132,290)(133,295)(134,296)(135,297)(136,321)
(137,319)(138,320)(139,316)(140,317)(141,318)(142,323)(143,324)(144,322)
(145,311)(146,312)(147,310)(148,309)(149,307)(150,308)(151,313)(152,314)
(153,315)(154,301)(155,302)(156,303)(157,299)(158,300)(159,298)(160,306)
(161,304)(162,305)(163,410)(164,411)(165,409)(166,408)(167,406)(168,407)
(169,412)(170,413)(171,414)(172,427)(173,428)(174,429)(175,425)(176,426)
(177,424)(178,432)(179,430)(180,431)(181,420)(182,418)(183,419)(184,415)
(185,416)(186,417)(187,422)(188,423)(189,421)(190,449)(191,450)(192,448)
(193,447)(194,445)(195,446)(196,442)(197,443)(198,444)(199,439)(200,440)
(201,441)(202,437)(203,438)(204,436)(205,435)(206,433)(207,434)(208,459)
(209,457)(210,458)(211,454)(212,455)(213,456)(214,452)(215,453)(216,451)
(217,484)(218,485)(219,486)(220,482)(221,483)(222,481)(223,480)(224,478)
(225,479)(226,477)(227,475)(228,476)(229,472)(230,473)(231,474)(232,470)
(233,471)(234,469)(235,467)(236,468)(237,466)(238,465)(239,463)(240,464)
(241,460)(242,461)(243,462);;
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*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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, 57)( 29, 56)( 30, 55)
( 31, 59)( 32, 58)( 33, 60)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)
( 39, 73)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)
( 47, 65)( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)
( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)
( 96,103)( 97,107)( 98,106)( 99,108)(109,138)(110,137)(111,136)(112,140)
(113,139)(114,141)(115,142)(116,144)(117,143)(118,156)(119,155)(120,154)
(121,158)(122,157)(123,159)(124,160)(125,162)(126,161)(127,147)(128,146)
(129,145)(130,149)(131,148)(132,150)(133,151)(134,153)(135,152)(164,165)
(166,168)(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)
(178,188)(179,187)(180,189)(190,219)(191,218)(192,217)(193,221)(194,220)
(195,222)(196,223)(197,225)(198,224)(199,237)(200,236)(201,235)(202,239)
(203,238)(204,240)(205,241)(206,243)(207,242)(208,228)(209,227)(210,226)
(211,230)(212,229)(213,231)(214,232)(215,234)(216,233)(245,246)(247,249)
(250,251)(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)
(260,268)(261,270)(271,300)(272,299)(273,298)(274,302)(275,301)(276,303)
(277,304)(278,306)(279,305)(280,318)(281,317)(282,316)(283,320)(284,319)
(285,321)(286,322)(287,324)(288,323)(289,309)(290,308)(291,307)(292,311)
(293,310)(294,312)(295,313)(296,315)(297,314)(326,327)(328,330)(331,332)
(334,343)(335,345)(336,344)(337,348)(338,347)(339,346)(340,350)(341,349)
(342,351)(352,381)(353,380)(354,379)(355,383)(356,382)(357,384)(358,385)
(359,387)(360,386)(361,399)(362,398)(363,397)(364,401)(365,400)(366,402)
(367,403)(368,405)(369,404)(370,390)(371,389)(372,388)(373,392)(374,391)
(375,393)(376,394)(377,396)(378,395)(407,408)(409,411)(412,413)(415,424)
(416,426)(417,425)(418,429)(419,428)(420,427)(421,431)(422,430)(423,432)
(433,462)(434,461)(435,460)(436,464)(437,463)(438,465)(439,466)(440,468)
(441,467)(442,480)(443,479)(444,478)(445,482)(446,481)(447,483)(448,484)
(449,486)(450,485)(451,471)(452,470)(453,469)(454,473)(455,472)(456,474)
(457,475)(458,477)(459,476);
s1 := Sym(486)!( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)
( 8, 33)( 9, 32)( 10, 45)( 11, 44)( 12, 43)( 13, 42)( 14, 41)( 15, 40)
( 16, 39)( 17, 38)( 18, 37)( 19, 50)( 20, 49)( 21, 51)( 22, 47)( 23, 46)
( 24, 48)( 25, 53)( 26, 52)( 27, 54)( 55, 57)( 58, 63)( 59, 62)( 60, 61)
( 64, 71)( 65, 70)( 66, 72)( 67, 68)( 73, 76)( 74, 78)( 75, 77)( 80, 81)
( 82,194)( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)( 88,197)( 89,196)
( 90,198)( 91,199)( 92,201)( 93,200)( 94,205)( 95,207)( 96,206)( 97,202)
( 98,204)( 99,203)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)
(106,210)(107,209)(108,208)(109,167)(110,166)(111,168)(112,164)(113,163)
(114,165)(115,170)(116,169)(117,171)(118,172)(119,174)(120,173)(121,178)
(122,180)(123,179)(124,175)(125,177)(126,176)(127,189)(128,188)(129,187)
(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,220)(137,222)
(138,221)(139,217)(140,219)(141,218)(142,223)(143,225)(144,224)(145,228)
(146,227)(147,226)(148,234)(149,233)(150,232)(151,231)(152,230)(153,229)
(154,242)(155,241)(156,243)(157,239)(158,238)(159,240)(160,236)(161,235)
(162,237)(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)(250,274)
(251,276)(252,275)(253,288)(254,287)(255,286)(256,285)(257,284)(258,283)
(259,282)(260,281)(261,280)(262,293)(263,292)(264,294)(265,290)(266,289)
(267,291)(268,296)(269,295)(270,297)(298,300)(301,306)(302,305)(303,304)
(307,314)(308,313)(309,315)(310,311)(316,319)(317,321)(318,320)(323,324)
(325,437)(326,436)(327,438)(328,434)(329,433)(330,435)(331,440)(332,439)
(333,441)(334,442)(335,444)(336,443)(337,448)(338,450)(339,449)(340,445)
(341,447)(342,446)(343,459)(344,458)(345,457)(346,456)(347,455)(348,454)
(349,453)(350,452)(351,451)(352,410)(353,409)(354,411)(355,407)(356,406)
(357,408)(358,413)(359,412)(360,414)(361,415)(362,417)(363,416)(364,421)
(365,423)(366,422)(367,418)(368,420)(369,419)(370,432)(371,431)(372,430)
(373,429)(374,428)(375,427)(376,426)(377,425)(378,424)(379,463)(380,465)
(381,464)(382,460)(383,462)(384,461)(385,466)(386,468)(387,467)(388,471)
(389,470)(390,469)(391,477)(392,476)(393,475)(394,474)(395,473)(396,472)
(397,485)(398,484)(399,486)(400,482)(401,481)(402,483)(403,479)(404,478)
(405,480);
s2 := Sym(486)!( 1,325)( 2,326)( 3,327)( 4,332)( 5,333)( 6,331)( 7,330)
( 8,328)( 9,329)( 10,345)( 11,343)( 12,344)( 13,349)( 14,350)( 15,351)
( 16,347)( 17,348)( 18,346)( 19,335)( 20,336)( 21,334)( 22,342)( 23,340)
( 24,341)( 25,337)( 26,338)( 27,339)( 28,364)( 29,365)( 30,366)( 31,362)
( 32,363)( 33,361)( 34,369)( 35,367)( 36,368)( 37,357)( 38,355)( 39,356)
( 40,352)( 41,353)( 42,354)( 43,359)( 44,360)( 45,358)( 46,374)( 47,375)
( 48,373)( 49,372)( 50,370)( 51,371)( 52,376)( 53,377)( 54,378)( 55,402)
( 56,400)( 57,401)( 58,397)( 59,398)( 60,399)( 61,404)( 62,405)( 63,403)
( 64,392)( 65,393)( 66,391)( 67,390)( 68,388)( 69,389)( 70,394)( 71,395)
( 72,396)( 73,382)( 74,383)( 75,384)( 76,380)( 77,381)( 78,379)( 79,387)
( 80,385)( 81,386)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)
( 88,249)( 89,247)( 90,248)( 91,264)( 92,262)( 93,263)( 94,268)( 95,269)
( 96,270)( 97,266)( 98,267)( 99,265)(100,254)(101,255)(102,253)(103,261)
(104,259)(105,260)(106,256)(107,257)(108,258)(109,283)(110,284)(111,285)
(112,281)(113,282)(114,280)(115,288)(116,286)(117,287)(118,276)(119,274)
(120,275)(121,271)(122,272)(123,273)(124,278)(125,279)(126,277)(127,293)
(128,294)(129,292)(130,291)(131,289)(132,290)(133,295)(134,296)(135,297)
(136,321)(137,319)(138,320)(139,316)(140,317)(141,318)(142,323)(143,324)
(144,322)(145,311)(146,312)(147,310)(148,309)(149,307)(150,308)(151,313)
(152,314)(153,315)(154,301)(155,302)(156,303)(157,299)(158,300)(159,298)
(160,306)(161,304)(162,305)(163,410)(164,411)(165,409)(166,408)(167,406)
(168,407)(169,412)(170,413)(171,414)(172,427)(173,428)(174,429)(175,425)
(176,426)(177,424)(178,432)(179,430)(180,431)(181,420)(182,418)(183,419)
(184,415)(185,416)(186,417)(187,422)(188,423)(189,421)(190,449)(191,450)
(192,448)(193,447)(194,445)(195,446)(196,442)(197,443)(198,444)(199,439)
(200,440)(201,441)(202,437)(203,438)(204,436)(205,435)(206,433)(207,434)
(208,459)(209,457)(210,458)(211,454)(212,455)(213,456)(214,452)(215,453)
(216,451)(217,484)(218,485)(219,486)(220,482)(221,483)(222,481)(223,480)
(224,478)(225,479)(226,477)(227,475)(228,476)(229,472)(230,473)(231,474)
(232,470)(233,471)(234,469)(235,467)(236,468)(237,466)(238,465)(239,463)
(240,464)(241,460)(242,461)(243,462);
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*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope