Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944q
if this polytope has a name.
Group : SmallGroup(1944,949)
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 : {9,18}*972f
   3-fold quotients : {18,6}*648c
   6-fold quotients : {9,6}*324c
   9-fold quotients : {6,6}*216c
   18-fold quotients : {3,6}*108
   27-fold quotients : {6,6}*72c
   54-fold quotients : {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 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)
( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 82,172)( 83,173)( 84,174)( 85,178)
( 86,179)( 87,180)( 88,175)( 89,176)( 90,177)( 91,163)( 92,164)( 93,165)
( 94,169)( 95,170)( 96,171)( 97,166)( 98,167)( 99,168)(100,181)(101,182)
(102,183)(103,187)(104,188)(105,189)(106,184)(107,185)(108,186)(109,199)
(110,200)(111,201)(112,205)(113,206)(114,207)(115,202)(116,203)(117,204)
(118,190)(119,191)(120,192)(121,196)(122,197)(123,198)(124,193)(125,194)
(126,195)(127,208)(128,209)(129,210)(130,214)(131,215)(132,216)(133,211)
(134,212)(135,213)(136,226)(137,227)(138,228)(139,232)(140,233)(141,234)
(142,229)(143,230)(144,231)(145,217)(146,218)(147,219)(148,223)(149,224)
(150,225)(151,220)(152,221)(153,222)(154,235)(155,236)(156,237)(157,241)
(158,242)(159,243)(160,238)(161,239)(162,240)(247,250)(248,251)(249,252)
(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)(260,266)
(261,267)(274,277)(275,278)(276,279)(280,289)(281,290)(282,291)(283,295)
(284,296)(285,297)(286,292)(287,293)(288,294)(301,304)(302,305)(303,306)
(307,316)(308,317)(309,318)(310,322)(311,323)(312,324)(313,319)(314,320)
(315,321)(325,415)(326,416)(327,417)(328,421)(329,422)(330,423)(331,418)
(332,419)(333,420)(334,406)(335,407)(336,408)(337,412)(338,413)(339,414)
(340,409)(341,410)(342,411)(343,424)(344,425)(345,426)(346,430)(347,431)
(348,432)(349,427)(350,428)(351,429)(352,442)(353,443)(354,444)(355,448)
(356,449)(357,450)(358,445)(359,446)(360,447)(361,433)(362,434)(363,435)
(364,439)(365,440)(366,441)(367,436)(368,437)(369,438)(370,451)(371,452)
(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)(379,469)
(380,470)(381,471)(382,475)(383,476)(384,477)(385,472)(386,473)(387,474)
(388,460)(389,461)(390,462)(391,466)(392,467)(393,468)(394,463)(395,464)
(396,465)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)(403,481)
(404,482)(405,483);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*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*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)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)
( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 82,172)( 83,173)( 84,174)
( 85,178)( 86,179)( 87,180)( 88,175)( 89,176)( 90,177)( 91,163)( 92,164)
( 93,165)( 94,169)( 95,170)( 96,171)( 97,166)( 98,167)( 99,168)(100,181)
(101,182)(102,183)(103,187)(104,188)(105,189)(106,184)(107,185)(108,186)
(109,199)(110,200)(111,201)(112,205)(113,206)(114,207)(115,202)(116,203)
(117,204)(118,190)(119,191)(120,192)(121,196)(122,197)(123,198)(124,193)
(125,194)(126,195)(127,208)(128,209)(129,210)(130,214)(131,215)(132,216)
(133,211)(134,212)(135,213)(136,226)(137,227)(138,228)(139,232)(140,233)
(141,234)(142,229)(143,230)(144,231)(145,217)(146,218)(147,219)(148,223)
(149,224)(150,225)(151,220)(152,221)(153,222)(154,235)(155,236)(156,237)
(157,241)(158,242)(159,243)(160,238)(161,239)(162,240)(247,250)(248,251)
(249,252)(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)
(260,266)(261,267)(274,277)(275,278)(276,279)(280,289)(281,290)(282,291)
(283,295)(284,296)(285,297)(286,292)(287,293)(288,294)(301,304)(302,305)
(303,306)(307,316)(308,317)(309,318)(310,322)(311,323)(312,324)(313,319)
(314,320)(315,321)(325,415)(326,416)(327,417)(328,421)(329,422)(330,423)
(331,418)(332,419)(333,420)(334,406)(335,407)(336,408)(337,412)(338,413)
(339,414)(340,409)(341,410)(342,411)(343,424)(344,425)(345,426)(346,430)
(347,431)(348,432)(349,427)(350,428)(351,429)(352,442)(353,443)(354,444)
(355,448)(356,449)(357,450)(358,445)(359,446)(360,447)(361,433)(362,434)
(363,435)(364,439)(365,440)(366,441)(367,436)(368,437)(369,438)(370,451)
(371,452)(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)
(379,469)(380,470)(381,471)(382,475)(383,476)(384,477)(385,472)(386,473)
(387,474)(388,460)(389,461)(390,462)(391,466)(392,467)(393,468)(394,463)
(395,464)(396,465)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)
(403,481)(404,482)(405,483);
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*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*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