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