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