Polytope of Type {6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1944l
if this polytope has a name.
Group : SmallGroup(1944,956)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 54, 486, 162
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 : {6,18}*972e
3-fold quotients : {6,18}*648g
6-fold quotients : {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)( 5, 6)( 8, 9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)( 14, 22)
( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)
( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 74)( 38, 73)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 65)( 47, 64)
( 48, 66)( 49, 68)( 50, 67)( 51, 69)( 52, 71)( 53, 70)( 54, 72)( 83, 84)
( 86, 87)( 89, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 96,105)
( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,139)(113,141)
(114,140)(115,142)(116,144)(117,143)(118,155)(119,154)(120,156)(121,158)
(122,157)(123,159)(124,161)(125,160)(126,162)(127,146)(128,145)(129,147)
(130,149)(131,148)(132,150)(133,152)(134,151)(135,153)(164,165)(167,168)
(170,171)(172,182)(173,181)(174,183)(175,185)(176,184)(177,186)(178,188)
(179,187)(180,189)(190,217)(191,219)(192,218)(193,220)(194,222)(195,221)
(196,223)(197,225)(198,224)(199,236)(200,235)(201,237)(202,239)(203,238)
(204,240)(205,242)(206,241)(207,243)(208,227)(209,226)(210,228)(211,230)
(212,229)(213,231)(214,233)(215,232)(216,234)(245,246)(248,249)(251,252)
(253,263)(254,262)(255,264)(256,266)(257,265)(258,267)(259,269)(260,268)
(261,270)(271,298)(272,300)(273,299)(274,301)(275,303)(276,302)(277,304)
(278,306)(279,305)(280,317)(281,316)(282,318)(283,320)(284,319)(285,321)
(286,323)(287,322)(288,324)(289,308)(290,307)(291,309)(292,311)(293,310)
(294,312)(295,314)(296,313)(297,315)(326,327)(329,330)(332,333)(334,344)
(335,343)(336,345)(337,347)(338,346)(339,348)(340,350)(341,349)(342,351)
(352,379)(353,381)(354,380)(355,382)(356,384)(357,383)(358,385)(359,387)
(360,386)(361,398)(362,397)(363,399)(364,401)(365,400)(366,402)(367,404)
(368,403)(369,405)(370,389)(371,388)(372,390)(373,392)(374,391)(375,393)
(376,395)(377,394)(378,396)(407,408)(410,411)(413,414)(415,425)(416,424)
(417,426)(418,428)(419,427)(420,429)(421,431)(422,430)(423,432)(433,460)
(434,462)(435,461)(436,463)(437,465)(438,464)(439,466)(440,468)(441,467)
(442,479)(443,478)(444,480)(445,482)(446,481)(447,483)(448,485)(449,484)
(450,486)(451,470)(452,469)(453,471)(454,473)(455,472)(456,474)(457,476)
(458,475)(459,477);;
s1 := ( 1, 28)( 2, 29)( 3, 30)( 4, 35)( 5, 36)( 6, 34)( 7, 33)( 8, 31)
( 9, 32)( 10, 45)( 11, 43)( 12, 44)( 13, 40)( 14, 41)( 15, 42)( 16, 38)
( 17, 39)( 18, 37)( 19, 50)( 20, 51)( 21, 49)( 22, 48)( 23, 46)( 24, 47)
( 25, 52)( 26, 53)( 27, 54)( 58, 62)( 59, 63)( 60, 61)( 64, 72)( 65, 70)
( 66, 71)( 73, 77)( 74, 78)( 75, 76)( 82,195)( 83,193)( 84,194)( 85,190)
( 86,191)( 87,192)( 88,197)( 89,198)( 90,196)( 91,200)( 92,201)( 93,199)
( 94,207)( 95,205)( 96,206)( 97,202)( 98,203)( 99,204)(100,214)(101,215)
(102,216)(103,212)(104,213)(105,211)(106,210)(107,208)(108,209)(109,166)
(110,167)(111,168)(112,164)(113,165)(114,163)(115,171)(116,169)(117,170)
(118,174)(119,172)(120,173)(121,178)(122,179)(123,180)(124,176)(125,177)
(126,175)(127,188)(128,189)(129,187)(130,186)(131,184)(132,185)(133,181)
(134,182)(135,183)(136,221)(137,222)(138,220)(139,219)(140,217)(141,218)
(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,233)(149,234)
(150,232)(151,231)(152,229)(153,230)(154,243)(155,241)(156,242)(157,238)
(158,239)(159,240)(160,236)(161,237)(162,235)(244,271)(245,272)(246,273)
(247,278)(248,279)(249,277)(250,276)(251,274)(252,275)(253,288)(254,286)
(255,287)(256,283)(257,284)(258,285)(259,281)(260,282)(261,280)(262,293)
(263,294)(264,292)(265,291)(266,289)(267,290)(268,295)(269,296)(270,297)
(301,305)(302,306)(303,304)(307,315)(308,313)(309,314)(316,320)(317,321)
(318,319)(325,438)(326,436)(327,437)(328,433)(329,434)(330,435)(331,440)
(332,441)(333,439)(334,443)(335,444)(336,442)(337,450)(338,448)(339,449)
(340,445)(341,446)(342,447)(343,457)(344,458)(345,459)(346,455)(347,456)
(348,454)(349,453)(350,451)(351,452)(352,409)(353,410)(354,411)(355,407)
(356,408)(357,406)(358,414)(359,412)(360,413)(361,417)(362,415)(363,416)
(364,421)(365,422)(366,423)(367,419)(368,420)(369,418)(370,431)(371,432)
(372,430)(373,429)(374,427)(375,428)(376,424)(377,425)(378,426)(379,464)
(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)(387,468)
(388,469)(389,470)(390,471)(391,476)(392,477)(393,475)(394,474)(395,472)
(396,473)(397,486)(398,484)(399,485)(400,481)(401,482)(402,483)(403,479)
(404,480)(405,478);;
s2 := ( 1,325)( 2,327)( 3,326)( 4,331)( 5,333)( 6,332)( 7,328)( 8,330)
( 9,329)( 10,344)( 11,343)( 12,345)( 13,350)( 14,349)( 15,351)( 16,347)
( 17,346)( 18,348)( 19,335)( 20,334)( 21,336)( 22,341)( 23,340)( 24,342)
( 25,338)( 26,337)( 27,339)( 28,364)( 29,366)( 30,365)( 31,361)( 32,363)
( 33,362)( 34,367)( 35,369)( 36,368)( 37,355)( 38,357)( 39,356)( 40,352)
( 41,354)( 42,353)( 43,358)( 44,360)( 45,359)( 46,374)( 47,373)( 48,375)
( 49,371)( 50,370)( 51,372)( 52,377)( 53,376)( 54,378)( 55,401)( 56,400)
( 57,402)( 58,398)( 59,397)( 60,399)( 61,404)( 62,403)( 63,405)( 64,392)
( 65,391)( 66,393)( 67,389)( 68,388)( 69,390)( 70,395)( 71,394)( 72,396)
( 73,383)( 74,382)( 75,384)( 76,380)( 77,379)( 78,381)( 79,386)( 80,385)
( 81,387)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)( 88,247)
( 89,249)( 90,248)( 91,263)( 92,262)( 93,264)( 94,269)( 95,268)( 96,270)
( 97,266)( 98,265)( 99,267)(100,254)(101,253)(102,255)(103,260)(104,259)
(105,261)(106,257)(107,256)(108,258)(109,283)(110,285)(111,284)(112,280)
(113,282)(114,281)(115,286)(116,288)(117,287)(118,274)(119,276)(120,275)
(121,271)(122,273)(123,272)(124,277)(125,279)(126,278)(127,293)(128,292)
(129,294)(130,290)(131,289)(132,291)(133,296)(134,295)(135,297)(136,320)
(137,319)(138,321)(139,317)(140,316)(141,318)(142,323)(143,322)(144,324)
(145,311)(146,310)(147,312)(148,308)(149,307)(150,309)(151,314)(152,313)
(153,315)(154,302)(155,301)(156,303)(157,299)(158,298)(159,300)(160,305)
(161,304)(162,306)(163,409)(164,411)(165,410)(166,406)(167,408)(168,407)
(169,412)(170,414)(171,413)(172,428)(173,427)(174,429)(175,425)(176,424)
(177,426)(178,431)(179,430)(180,432)(181,419)(182,418)(183,420)(184,416)
(185,415)(186,417)(187,422)(188,421)(189,423)(190,450)(191,449)(192,448)
(193,447)(194,446)(195,445)(196,444)(197,443)(198,442)(199,441)(200,440)
(201,439)(202,438)(203,437)(204,436)(205,435)(206,434)(207,433)(208,457)
(209,459)(210,458)(211,454)(212,456)(213,455)(214,451)(215,453)(216,452)
(217,486)(218,485)(219,484)(220,483)(221,482)(222,481)(223,480)(224,479)
(225,478)(226,477)(227,476)(228,475)(229,474)(230,473)(231,472)(232,471)
(233,470)(234,469)(235,468)(236,467)(237,466)(238,465)(239,464)(240,463)
(241,462)(242,461)(243,460);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*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)( 5, 6)( 8, 9)( 10, 20)( 11, 19)( 12, 21)( 13, 23)
( 14, 22)( 15, 24)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)
( 31, 58)( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 74)( 38, 73)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 65)
( 47, 64)( 48, 66)( 49, 68)( 50, 67)( 51, 69)( 52, 71)( 53, 70)( 54, 72)
( 83, 84)( 86, 87)( 89, 90)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)
( 96,105)( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,139)
(113,141)(114,140)(115,142)(116,144)(117,143)(118,155)(119,154)(120,156)
(121,158)(122,157)(123,159)(124,161)(125,160)(126,162)(127,146)(128,145)
(129,147)(130,149)(131,148)(132,150)(133,152)(134,151)(135,153)(164,165)
(167,168)(170,171)(172,182)(173,181)(174,183)(175,185)(176,184)(177,186)
(178,188)(179,187)(180,189)(190,217)(191,219)(192,218)(193,220)(194,222)
(195,221)(196,223)(197,225)(198,224)(199,236)(200,235)(201,237)(202,239)
(203,238)(204,240)(205,242)(206,241)(207,243)(208,227)(209,226)(210,228)
(211,230)(212,229)(213,231)(214,233)(215,232)(216,234)(245,246)(248,249)
(251,252)(253,263)(254,262)(255,264)(256,266)(257,265)(258,267)(259,269)
(260,268)(261,270)(271,298)(272,300)(273,299)(274,301)(275,303)(276,302)
(277,304)(278,306)(279,305)(280,317)(281,316)(282,318)(283,320)(284,319)
(285,321)(286,323)(287,322)(288,324)(289,308)(290,307)(291,309)(292,311)
(293,310)(294,312)(295,314)(296,313)(297,315)(326,327)(329,330)(332,333)
(334,344)(335,343)(336,345)(337,347)(338,346)(339,348)(340,350)(341,349)
(342,351)(352,379)(353,381)(354,380)(355,382)(356,384)(357,383)(358,385)
(359,387)(360,386)(361,398)(362,397)(363,399)(364,401)(365,400)(366,402)
(367,404)(368,403)(369,405)(370,389)(371,388)(372,390)(373,392)(374,391)
(375,393)(376,395)(377,394)(378,396)(407,408)(410,411)(413,414)(415,425)
(416,424)(417,426)(418,428)(419,427)(420,429)(421,431)(422,430)(423,432)
(433,460)(434,462)(435,461)(436,463)(437,465)(438,464)(439,466)(440,468)
(441,467)(442,479)(443,478)(444,480)(445,482)(446,481)(447,483)(448,485)
(449,484)(450,486)(451,470)(452,469)(453,471)(454,473)(455,472)(456,474)
(457,476)(458,475)(459,477);
s1 := Sym(486)!( 1, 28)( 2, 29)( 3, 30)( 4, 35)( 5, 36)( 6, 34)( 7, 33)
( 8, 31)( 9, 32)( 10, 45)( 11, 43)( 12, 44)( 13, 40)( 14, 41)( 15, 42)
( 16, 38)( 17, 39)( 18, 37)( 19, 50)( 20, 51)( 21, 49)( 22, 48)( 23, 46)
( 24, 47)( 25, 52)( 26, 53)( 27, 54)( 58, 62)( 59, 63)( 60, 61)( 64, 72)
( 65, 70)( 66, 71)( 73, 77)( 74, 78)( 75, 76)( 82,195)( 83,193)( 84,194)
( 85,190)( 86,191)( 87,192)( 88,197)( 89,198)( 90,196)( 91,200)( 92,201)
( 93,199)( 94,207)( 95,205)( 96,206)( 97,202)( 98,203)( 99,204)(100,214)
(101,215)(102,216)(103,212)(104,213)(105,211)(106,210)(107,208)(108,209)
(109,166)(110,167)(111,168)(112,164)(113,165)(114,163)(115,171)(116,169)
(117,170)(118,174)(119,172)(120,173)(121,178)(122,179)(123,180)(124,176)
(125,177)(126,175)(127,188)(128,189)(129,187)(130,186)(131,184)(132,185)
(133,181)(134,182)(135,183)(136,221)(137,222)(138,220)(139,219)(140,217)
(141,218)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,233)
(149,234)(150,232)(151,231)(152,229)(153,230)(154,243)(155,241)(156,242)
(157,238)(158,239)(159,240)(160,236)(161,237)(162,235)(244,271)(245,272)
(246,273)(247,278)(248,279)(249,277)(250,276)(251,274)(252,275)(253,288)
(254,286)(255,287)(256,283)(257,284)(258,285)(259,281)(260,282)(261,280)
(262,293)(263,294)(264,292)(265,291)(266,289)(267,290)(268,295)(269,296)
(270,297)(301,305)(302,306)(303,304)(307,315)(308,313)(309,314)(316,320)
(317,321)(318,319)(325,438)(326,436)(327,437)(328,433)(329,434)(330,435)
(331,440)(332,441)(333,439)(334,443)(335,444)(336,442)(337,450)(338,448)
(339,449)(340,445)(341,446)(342,447)(343,457)(344,458)(345,459)(346,455)
(347,456)(348,454)(349,453)(350,451)(351,452)(352,409)(353,410)(354,411)
(355,407)(356,408)(357,406)(358,414)(359,412)(360,413)(361,417)(362,415)
(363,416)(364,421)(365,422)(366,423)(367,419)(368,420)(369,418)(370,431)
(371,432)(372,430)(373,429)(374,427)(375,428)(376,424)(377,425)(378,426)
(379,464)(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)
(387,468)(388,469)(389,470)(390,471)(391,476)(392,477)(393,475)(394,474)
(395,472)(396,473)(397,486)(398,484)(399,485)(400,481)(401,482)(402,483)
(403,479)(404,480)(405,478);
s2 := Sym(486)!( 1,325)( 2,327)( 3,326)( 4,331)( 5,333)( 6,332)( 7,328)
( 8,330)( 9,329)( 10,344)( 11,343)( 12,345)( 13,350)( 14,349)( 15,351)
( 16,347)( 17,346)( 18,348)( 19,335)( 20,334)( 21,336)( 22,341)( 23,340)
( 24,342)( 25,338)( 26,337)( 27,339)( 28,364)( 29,366)( 30,365)( 31,361)
( 32,363)( 33,362)( 34,367)( 35,369)( 36,368)( 37,355)( 38,357)( 39,356)
( 40,352)( 41,354)( 42,353)( 43,358)( 44,360)( 45,359)( 46,374)( 47,373)
( 48,375)( 49,371)( 50,370)( 51,372)( 52,377)( 53,376)( 54,378)( 55,401)
( 56,400)( 57,402)( 58,398)( 59,397)( 60,399)( 61,404)( 62,403)( 63,405)
( 64,392)( 65,391)( 66,393)( 67,389)( 68,388)( 69,390)( 70,395)( 71,394)
( 72,396)( 73,383)( 74,382)( 75,384)( 76,380)( 77,379)( 78,381)( 79,386)
( 80,385)( 81,387)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)
( 88,247)( 89,249)( 90,248)( 91,263)( 92,262)( 93,264)( 94,269)( 95,268)
( 96,270)( 97,266)( 98,265)( 99,267)(100,254)(101,253)(102,255)(103,260)
(104,259)(105,261)(106,257)(107,256)(108,258)(109,283)(110,285)(111,284)
(112,280)(113,282)(114,281)(115,286)(116,288)(117,287)(118,274)(119,276)
(120,275)(121,271)(122,273)(123,272)(124,277)(125,279)(126,278)(127,293)
(128,292)(129,294)(130,290)(131,289)(132,291)(133,296)(134,295)(135,297)
(136,320)(137,319)(138,321)(139,317)(140,316)(141,318)(142,323)(143,322)
(144,324)(145,311)(146,310)(147,312)(148,308)(149,307)(150,309)(151,314)
(152,313)(153,315)(154,302)(155,301)(156,303)(157,299)(158,298)(159,300)
(160,305)(161,304)(162,306)(163,409)(164,411)(165,410)(166,406)(167,408)
(168,407)(169,412)(170,414)(171,413)(172,428)(173,427)(174,429)(175,425)
(176,424)(177,426)(178,431)(179,430)(180,432)(181,419)(182,418)(183,420)
(184,416)(185,415)(186,417)(187,422)(188,421)(189,423)(190,450)(191,449)
(192,448)(193,447)(194,446)(195,445)(196,444)(197,443)(198,442)(199,441)
(200,440)(201,439)(202,438)(203,437)(204,436)(205,435)(206,434)(207,433)
(208,457)(209,459)(210,458)(211,454)(212,456)(213,455)(214,451)(215,453)
(216,452)(217,486)(218,485)(219,484)(220,483)(221,482)(222,481)(223,480)
(224,479)(225,478)(226,477)(227,476)(228,475)(229,474)(230,473)(231,472)
(232,471)(233,470)(234,469)(235,468)(236,467)(237,466)(238,465)(239,464)
(240,463)(241,462)(242,461)(243,460);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*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