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Polytope of Type {18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1944h
if this polytope has a name.
Group : SmallGroup(1944,949)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 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 : {9,6}*972d
   3-fold quotients : {18,6}*648a, {18,6}*648d
   6-fold quotients : {9,6}*324a, {9,6}*324b
   9-fold quotients : {18,6}*216b, {6,6}*216c
   18-fold quotients : {9,6}*108, {3,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72c
   54-fold quotients : {9,2}*36, {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)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 58)
( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 73)( 38, 75)( 39, 74)
( 40, 76)( 41, 78)( 42, 77)( 43, 79)( 44, 81)( 45, 80)( 46, 64)( 47, 66)
( 48, 65)( 49, 67)( 50, 69)( 51, 68)( 52, 70)( 53, 72)( 54, 71)( 82,172)
( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)( 89,180)( 90,179)
( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167)( 97,169)( 98,171)
( 99,170)(100,181)(101,183)(102,182)(103,184)(104,186)(105,185)(106,187)
(107,189)(108,188)(109,226)(110,228)(111,227)(112,229)(113,231)(114,230)
(115,232)(116,234)(117,233)(118,217)(119,219)(120,218)(121,220)(122,222)
(123,221)(124,223)(125,225)(126,224)(127,235)(128,237)(129,236)(130,238)
(131,240)(132,239)(133,241)(134,243)(135,242)(136,199)(137,201)(138,200)
(139,202)(140,204)(141,203)(142,205)(143,207)(144,206)(145,190)(146,192)
(147,191)(148,193)(149,195)(150,194)(151,196)(152,198)(153,197)(154,208)
(155,210)(156,209)(157,211)(158,213)(159,212)(160,214)(161,216)(162,215)
(245,246)(248,249)(251,252)(253,262)(254,264)(255,263)(256,265)(257,267)
(258,266)(259,268)(260,270)(261,269)(271,298)(272,300)(273,299)(274,301)
(275,303)(276,302)(277,304)(278,306)(279,305)(280,316)(281,318)(282,317)
(283,319)(284,321)(285,320)(286,322)(287,324)(288,323)(289,307)(290,309)
(291,308)(292,310)(293,312)(294,311)(295,313)(296,315)(297,314)(325,415)
(326,417)(327,416)(328,418)(329,420)(330,419)(331,421)(332,423)(333,422)
(334,406)(335,408)(336,407)(337,409)(338,411)(339,410)(340,412)(341,414)
(342,413)(343,424)(344,426)(345,425)(346,427)(347,429)(348,428)(349,430)
(350,432)(351,431)(352,469)(353,471)(354,470)(355,472)(356,474)(357,473)
(358,475)(359,477)(360,476)(361,460)(362,462)(363,461)(364,463)(365,465)
(366,464)(367,466)(368,468)(369,467)(370,478)(371,480)(372,479)(373,481)
(374,483)(375,482)(376,484)(377,486)(378,485)(379,442)(380,444)(381,443)
(382,445)(383,447)(384,446)(385,448)(386,450)(387,449)(388,433)(389,435)
(390,434)(391,436)(392,438)(393,437)(394,439)(395,441)(396,440)(397,451)
(398,453)(399,452)(400,454)(401,456)(402,455)(403,457)(404,459)(405,458);;
s1 := (  1,436)(  2,438)(  3,437)(  4,440)(  5,439)(  6,441)(  7,435)(  8,434)
(  9,433)( 10,454)( 11,456)( 12,455)( 13,458)( 14,457)( 15,459)( 16,453)
( 17,452)( 18,451)( 19,445)( 20,447)( 21,446)( 22,449)( 23,448)( 24,450)
( 25,444)( 26,443)( 27,442)( 28,414)( 29,413)( 30,412)( 31,406)( 32,408)
( 33,407)( 34,410)( 35,409)( 36,411)( 37,432)( 38,431)( 39,430)( 40,424)
( 41,426)( 42,425)( 43,428)( 44,427)( 45,429)( 46,423)( 47,422)( 48,421)
( 49,415)( 50,417)( 51,416)( 52,419)( 53,418)( 54,420)( 55,461)( 56,460)
( 57,462)( 58,465)( 59,464)( 60,463)( 61,466)( 62,468)( 63,467)( 64,479)
( 65,478)( 66,480)( 67,483)( 68,482)( 69,481)( 70,484)( 71,486)( 72,485)
( 73,470)( 74,469)( 75,471)( 76,474)( 77,473)( 78,472)( 79,475)( 80,477)
( 81,476)( 82,355)( 83,357)( 84,356)( 85,359)( 86,358)( 87,360)( 88,354)
( 89,353)( 90,352)( 91,373)( 92,375)( 93,374)( 94,377)( 95,376)( 96,378)
( 97,372)( 98,371)( 99,370)(100,364)(101,366)(102,365)(103,368)(104,367)
(105,369)(106,363)(107,362)(108,361)(109,333)(110,332)(111,331)(112,325)
(113,327)(114,326)(115,329)(116,328)(117,330)(118,351)(119,350)(120,349)
(121,343)(122,345)(123,344)(124,347)(125,346)(126,348)(127,342)(128,341)
(129,340)(130,334)(131,336)(132,335)(133,338)(134,337)(135,339)(136,380)
(137,379)(138,381)(139,384)(140,383)(141,382)(142,385)(143,387)(144,386)
(145,398)(146,397)(147,399)(148,402)(149,401)(150,400)(151,403)(152,405)
(153,404)(154,389)(155,388)(156,390)(157,393)(158,392)(159,391)(160,394)
(161,396)(162,395)(163,274)(164,276)(165,275)(166,278)(167,277)(168,279)
(169,273)(170,272)(171,271)(172,292)(173,294)(174,293)(175,296)(176,295)
(177,297)(178,291)(179,290)(180,289)(181,283)(182,285)(183,284)(184,287)
(185,286)(186,288)(187,282)(188,281)(189,280)(190,252)(191,251)(192,250)
(193,244)(194,246)(195,245)(196,248)(197,247)(198,249)(199,270)(200,269)
(201,268)(202,262)(203,264)(204,263)(205,266)(206,265)(207,267)(208,261)
(209,260)(210,259)(211,253)(212,255)(213,254)(214,257)(215,256)(216,258)
(217,299)(218,298)(219,300)(220,303)(221,302)(222,301)(223,304)(224,306)
(225,305)(226,317)(227,316)(228,318)(229,321)(230,320)(231,319)(232,322)
(233,324)(234,323)(235,308)(236,307)(237,309)(238,312)(239,311)(240,310)
(241,313)(242,315)(243,314);;
s2 := (  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, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)
(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)
(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)
(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)(277,301)
(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)(285,314)
(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)(293,324)
(294,323)(295,319)(296,321)(297,320)(326,327)(328,331)(329,333)(330,332)
(335,336)(337,340)(338,342)(339,341)(344,345)(346,349)(347,351)(348,350)
(352,379)(353,381)(354,380)(355,385)(356,387)(357,386)(358,382)(359,384)
(360,383)(361,388)(362,390)(363,389)(364,394)(365,396)(366,395)(367,391)
(368,393)(369,392)(370,397)(371,399)(372,398)(373,403)(374,405)(375,404)
(376,400)(377,402)(378,401)(407,408)(409,412)(410,414)(411,413)(416,417)
(418,421)(419,423)(420,422)(425,426)(427,430)(428,432)(429,431)(433,460)
(434,462)(435,461)(436,466)(437,468)(438,467)(439,463)(440,465)(441,464)
(442,469)(443,471)(444,470)(445,475)(446,477)(447,476)(448,472)(449,474)
(450,473)(451,478)(452,480)(453,479)(454,484)(455,486)(456,485)(457,481)
(458,483)(459,482);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(486)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 28, 55)( 29, 57)( 30, 56)
( 31, 58)( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 73)( 38, 75)
( 39, 74)( 40, 76)( 41, 78)( 42, 77)( 43, 79)( 44, 81)( 45, 80)( 46, 64)
( 47, 66)( 48, 65)( 49, 67)( 50, 69)( 51, 68)( 52, 70)( 53, 72)( 54, 71)
( 82,172)( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)( 89,180)
( 90,179)( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167)( 97,169)
( 98,171)( 99,170)(100,181)(101,183)(102,182)(103,184)(104,186)(105,185)
(106,187)(107,189)(108,188)(109,226)(110,228)(111,227)(112,229)(113,231)
(114,230)(115,232)(116,234)(117,233)(118,217)(119,219)(120,218)(121,220)
(122,222)(123,221)(124,223)(125,225)(126,224)(127,235)(128,237)(129,236)
(130,238)(131,240)(132,239)(133,241)(134,243)(135,242)(136,199)(137,201)
(138,200)(139,202)(140,204)(141,203)(142,205)(143,207)(144,206)(145,190)
(146,192)(147,191)(148,193)(149,195)(150,194)(151,196)(152,198)(153,197)
(154,208)(155,210)(156,209)(157,211)(158,213)(159,212)(160,214)(161,216)
(162,215)(245,246)(248,249)(251,252)(253,262)(254,264)(255,263)(256,265)
(257,267)(258,266)(259,268)(260,270)(261,269)(271,298)(272,300)(273,299)
(274,301)(275,303)(276,302)(277,304)(278,306)(279,305)(280,316)(281,318)
(282,317)(283,319)(284,321)(285,320)(286,322)(287,324)(288,323)(289,307)
(290,309)(291,308)(292,310)(293,312)(294,311)(295,313)(296,315)(297,314)
(325,415)(326,417)(327,416)(328,418)(329,420)(330,419)(331,421)(332,423)
(333,422)(334,406)(335,408)(336,407)(337,409)(338,411)(339,410)(340,412)
(341,414)(342,413)(343,424)(344,426)(345,425)(346,427)(347,429)(348,428)
(349,430)(350,432)(351,431)(352,469)(353,471)(354,470)(355,472)(356,474)
(357,473)(358,475)(359,477)(360,476)(361,460)(362,462)(363,461)(364,463)
(365,465)(366,464)(367,466)(368,468)(369,467)(370,478)(371,480)(372,479)
(373,481)(374,483)(375,482)(376,484)(377,486)(378,485)(379,442)(380,444)
(381,443)(382,445)(383,447)(384,446)(385,448)(386,450)(387,449)(388,433)
(389,435)(390,434)(391,436)(392,438)(393,437)(394,439)(395,441)(396,440)
(397,451)(398,453)(399,452)(400,454)(401,456)(402,455)(403,457)(404,459)
(405,458);
s1 := Sym(486)!(  1,436)(  2,438)(  3,437)(  4,440)(  5,439)(  6,441)(  7,435)
(  8,434)(  9,433)( 10,454)( 11,456)( 12,455)( 13,458)( 14,457)( 15,459)
( 16,453)( 17,452)( 18,451)( 19,445)( 20,447)( 21,446)( 22,449)( 23,448)
( 24,450)( 25,444)( 26,443)( 27,442)( 28,414)( 29,413)( 30,412)( 31,406)
( 32,408)( 33,407)( 34,410)( 35,409)( 36,411)( 37,432)( 38,431)( 39,430)
( 40,424)( 41,426)( 42,425)( 43,428)( 44,427)( 45,429)( 46,423)( 47,422)
( 48,421)( 49,415)( 50,417)( 51,416)( 52,419)( 53,418)( 54,420)( 55,461)
( 56,460)( 57,462)( 58,465)( 59,464)( 60,463)( 61,466)( 62,468)( 63,467)
( 64,479)( 65,478)( 66,480)( 67,483)( 68,482)( 69,481)( 70,484)( 71,486)
( 72,485)( 73,470)( 74,469)( 75,471)( 76,474)( 77,473)( 78,472)( 79,475)
( 80,477)( 81,476)( 82,355)( 83,357)( 84,356)( 85,359)( 86,358)( 87,360)
( 88,354)( 89,353)( 90,352)( 91,373)( 92,375)( 93,374)( 94,377)( 95,376)
( 96,378)( 97,372)( 98,371)( 99,370)(100,364)(101,366)(102,365)(103,368)
(104,367)(105,369)(106,363)(107,362)(108,361)(109,333)(110,332)(111,331)
(112,325)(113,327)(114,326)(115,329)(116,328)(117,330)(118,351)(119,350)
(120,349)(121,343)(122,345)(123,344)(124,347)(125,346)(126,348)(127,342)
(128,341)(129,340)(130,334)(131,336)(132,335)(133,338)(134,337)(135,339)
(136,380)(137,379)(138,381)(139,384)(140,383)(141,382)(142,385)(143,387)
(144,386)(145,398)(146,397)(147,399)(148,402)(149,401)(150,400)(151,403)
(152,405)(153,404)(154,389)(155,388)(156,390)(157,393)(158,392)(159,391)
(160,394)(161,396)(162,395)(163,274)(164,276)(165,275)(166,278)(167,277)
(168,279)(169,273)(170,272)(171,271)(172,292)(173,294)(174,293)(175,296)
(176,295)(177,297)(178,291)(179,290)(180,289)(181,283)(182,285)(183,284)
(184,287)(185,286)(186,288)(187,282)(188,281)(189,280)(190,252)(191,251)
(192,250)(193,244)(194,246)(195,245)(196,248)(197,247)(198,249)(199,270)
(200,269)(201,268)(202,262)(203,264)(204,263)(205,266)(206,265)(207,267)
(208,261)(209,260)(210,259)(211,253)(212,255)(213,254)(214,257)(215,256)
(216,258)(217,299)(218,298)(219,300)(220,303)(221,302)(222,301)(223,304)
(224,306)(225,305)(226,317)(227,316)(228,318)(229,321)(230,320)(231,319)
(232,322)(233,324)(234,323)(235,308)(236,307)(237,309)(238,312)(239,311)
(240,310)(241,313)(242,315)(243,314);
s2 := 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, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)
(195,224)(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)
(203,234)(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)
(211,241)(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)
(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)
(266,270)(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)
(277,301)(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)
(285,314)(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)
(293,324)(294,323)(295,319)(296,321)(297,320)(326,327)(328,331)(329,333)
(330,332)(335,336)(337,340)(338,342)(339,341)(344,345)(346,349)(347,351)
(348,350)(352,379)(353,381)(354,380)(355,385)(356,387)(357,386)(358,382)
(359,384)(360,383)(361,388)(362,390)(363,389)(364,394)(365,396)(366,395)
(367,391)(368,393)(369,392)(370,397)(371,399)(372,398)(373,403)(374,405)
(375,404)(376,400)(377,402)(378,401)(407,408)(409,412)(410,414)(411,413)
(416,417)(418,421)(419,423)(420,422)(425,426)(427,430)(428,432)(429,431)
(433,460)(434,462)(435,461)(436,466)(437,468)(438,467)(439,463)(440,465)
(441,464)(442,469)(443,471)(444,470)(445,475)(446,477)(447,476)(448,472)
(449,474)(450,473)(451,478)(452,480)(453,479)(454,484)(455,486)(456,485)
(457,481)(458,483)(459,482);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope