Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944a
if this polytope has a name.
Group : SmallGroup(1944,940)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 18
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}*972a
   3-fold quotients : {18,18}*648c, {18,6}*648a
   6-fold quotients : {9,18}*324, {9,6}*324a
   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 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 28, 58)( 29, 59)( 30, 60)( 31, 55)
( 32, 56)( 33, 57)( 34, 61)( 35, 62)( 36, 63)( 37, 76)( 38, 77)( 39, 78)
( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)( 47, 68)
( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)( 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)(109,229)(110,230)(111,231)(112,226)(113,227)(114,228)
(115,232)(116,233)(117,234)(118,220)(119,221)(120,222)(121,217)(122,218)
(123,219)(124,223)(125,224)(126,225)(127,238)(128,239)(129,240)(130,235)
(131,236)(132,237)(133,241)(134,242)(135,243)(136,202)(137,203)(138,204)
(139,199)(140,200)(141,201)(142,205)(143,206)(144,207)(145,193)(146,194)
(147,195)(148,190)(149,191)(150,192)(151,196)(152,197)(153,198)(154,211)
(155,212)(156,213)(157,208)(158,209)(159,210)(160,214)(161,215)(162,216)
(247,250)(248,251)(249,252)(253,262)(254,263)(255,264)(256,268)(257,269)
(258,270)(259,265)(260,266)(261,267)(271,301)(272,302)(273,303)(274,298)
(275,299)(276,300)(277,304)(278,305)(279,306)(280,319)(281,320)(282,321)
(283,316)(284,317)(285,318)(286,322)(287,323)(288,324)(289,310)(290,311)
(291,312)(292,307)(293,308)(294,309)(295,313)(296,314)(297,315)(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)(352,472)(353,473)(354,474)(355,469)(356,470)(357,471)
(358,475)(359,476)(360,477)(361,463)(362,464)(363,465)(364,460)(365,461)
(366,462)(367,466)(368,467)(369,468)(370,481)(371,482)(372,483)(373,478)
(374,479)(375,480)(376,484)(377,485)(378,486)(379,445)(380,446)(381,447)
(382,442)(383,443)(384,444)(385,448)(386,449)(387,450)(388,436)(389,437)
(390,438)(391,433)(392,434)(393,435)(394,439)(395,440)(396,441)(397,454)
(398,455)(399,456)(400,451)(401,452)(402,453)(403,457)(404,458)(405,459);;
s1 := (  1,434)(  2,435)(  3,433)(  4,440)(  5,441)(  6,439)(  7,437)(  8,438)
(  9,436)( 10,452)( 11,453)( 12,451)( 13,458)( 14,459)( 15,457)( 16,455)
( 17,456)( 18,454)( 19,443)( 20,444)( 21,442)( 22,449)( 23,450)( 24,448)
( 25,446)( 26,447)( 27,445)( 28,408)( 29,406)( 30,407)( 31,414)( 32,412)
( 33,413)( 34,411)( 35,409)( 36,410)( 37,426)( 38,424)( 39,425)( 40,432)
( 41,430)( 42,431)( 43,429)( 44,427)( 45,428)( 46,417)( 47,415)( 48,416)
( 49,423)( 50,421)( 51,422)( 52,420)( 53,418)( 54,419)( 55,463)( 56,464)
( 57,465)( 58,460)( 59,461)( 60,462)( 61,466)( 62,467)( 63,468)( 64,481)
( 65,482)( 66,483)( 67,478)( 68,479)( 69,480)( 70,484)( 71,485)( 72,486)
( 73,472)( 74,473)( 75,474)( 76,469)( 77,470)( 78,471)( 79,475)( 80,476)
( 81,477)( 82,353)( 83,354)( 84,352)( 85,359)( 86,360)( 87,358)( 88,356)
( 89,357)( 90,355)( 91,371)( 92,372)( 93,370)( 94,377)( 95,378)( 96,376)
( 97,374)( 98,375)( 99,373)(100,362)(101,363)(102,361)(103,368)(104,369)
(105,367)(106,365)(107,366)(108,364)(109,327)(110,325)(111,326)(112,333)
(113,331)(114,332)(115,330)(116,328)(117,329)(118,345)(119,343)(120,344)
(121,351)(122,349)(123,350)(124,348)(125,346)(126,347)(127,336)(128,334)
(129,335)(130,342)(131,340)(132,341)(133,339)(134,337)(135,338)(136,382)
(137,383)(138,384)(139,379)(140,380)(141,381)(142,385)(143,386)(144,387)
(145,400)(146,401)(147,402)(148,397)(149,398)(150,399)(151,403)(152,404)
(153,405)(154,391)(155,392)(156,393)(157,388)(158,389)(159,390)(160,394)
(161,395)(162,396)(163,272)(164,273)(165,271)(166,278)(167,279)(168,277)
(169,275)(170,276)(171,274)(172,290)(173,291)(174,289)(175,296)(176,297)
(177,295)(178,293)(179,294)(180,292)(181,281)(182,282)(183,280)(184,287)
(185,288)(186,286)(187,284)(188,285)(189,283)(190,246)(191,244)(192,245)
(193,252)(194,250)(195,251)(196,249)(197,247)(198,248)(199,264)(200,262)
(201,263)(202,270)(203,268)(204,269)(205,267)(206,265)(207,266)(208,255)
(209,253)(210,254)(211,261)(212,259)(213,260)(214,258)(215,256)(216,257)
(217,301)(218,302)(219,303)(220,298)(221,299)(222,300)(223,304)(224,305)
(225,306)(226,319)(227,320)(228,321)(229,316)(230,317)(231,318)(232,322)
(233,323)(234,324)(235,310)(236,311)(237,312)(238,307)(239,308)(240,309)
(241,313)(242,314)(243,315);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 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)(109,199)
(110,201)(111,200)(112,202)(113,204)(114,203)(115,205)(116,207)(117,206)
(118,190)(119,192)(120,191)(121,193)(122,195)(123,194)(124,196)(125,198)
(126,197)(127,208)(128,210)(129,209)(130,211)(131,213)(132,212)(133,214)
(134,216)(135,215)(136,226)(137,228)(138,227)(139,229)(140,231)(141,230)
(142,232)(143,234)(144,233)(145,217)(146,219)(147,218)(148,220)(149,222)
(150,221)(151,223)(152,225)(153,224)(154,235)(155,237)(156,236)(157,238)
(158,240)(159,239)(160,241)(161,243)(162,242)(245,246)(248,249)(251,252)
(253,262)(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)(260,270)
(261,269)(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)(283,292)
(284,294)(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)(305,306)
(307,316)(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)
(315,323)(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)(352,442)(353,444)(354,443)(355,445)
(356,447)(357,446)(358,448)(359,450)(360,449)(361,433)(362,435)(363,434)
(364,436)(365,438)(366,437)(367,439)(368,441)(369,440)(370,451)(371,453)
(372,452)(373,454)(374,456)(375,455)(376,457)(377,459)(378,458)(379,469)
(380,471)(381,470)(382,472)(383,474)(384,473)(385,475)(386,477)(387,476)
(388,460)(389,462)(390,461)(391,463)(392,465)(393,464)(394,466)(395,468)
(396,467)(397,478)(398,480)(399,479)(400,481)(401,483)(402,482)(403,484)
(404,486)(405,485);;
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*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, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := 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)( 28, 58)( 29, 59)( 30, 60)
( 31, 55)( 32, 56)( 33, 57)( 34, 61)( 35, 62)( 36, 63)( 37, 76)( 38, 77)
( 39, 78)( 40, 73)( 41, 74)( 42, 75)( 43, 79)( 44, 80)( 45, 81)( 46, 67)
( 47, 68)( 48, 69)( 49, 64)( 50, 65)( 51, 66)( 52, 70)( 53, 71)( 54, 72)
( 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)(109,229)(110,230)(111,231)(112,226)(113,227)
(114,228)(115,232)(116,233)(117,234)(118,220)(119,221)(120,222)(121,217)
(122,218)(123,219)(124,223)(125,224)(126,225)(127,238)(128,239)(129,240)
(130,235)(131,236)(132,237)(133,241)(134,242)(135,243)(136,202)(137,203)
(138,204)(139,199)(140,200)(141,201)(142,205)(143,206)(144,207)(145,193)
(146,194)(147,195)(148,190)(149,191)(150,192)(151,196)(152,197)(153,198)
(154,211)(155,212)(156,213)(157,208)(158,209)(159,210)(160,214)(161,215)
(162,216)(247,250)(248,251)(249,252)(253,262)(254,263)(255,264)(256,268)
(257,269)(258,270)(259,265)(260,266)(261,267)(271,301)(272,302)(273,303)
(274,298)(275,299)(276,300)(277,304)(278,305)(279,306)(280,319)(281,320)
(282,321)(283,316)(284,317)(285,318)(286,322)(287,323)(288,324)(289,310)
(290,311)(291,312)(292,307)(293,308)(294,309)(295,313)(296,314)(297,315)
(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)(352,472)(353,473)(354,474)(355,469)(356,470)
(357,471)(358,475)(359,476)(360,477)(361,463)(362,464)(363,465)(364,460)
(365,461)(366,462)(367,466)(368,467)(369,468)(370,481)(371,482)(372,483)
(373,478)(374,479)(375,480)(376,484)(377,485)(378,486)(379,445)(380,446)
(381,447)(382,442)(383,443)(384,444)(385,448)(386,449)(387,450)(388,436)
(389,437)(390,438)(391,433)(392,434)(393,435)(394,439)(395,440)(396,441)
(397,454)(398,455)(399,456)(400,451)(401,452)(402,453)(403,457)(404,458)
(405,459);
s1 := Sym(486)!(  1,434)(  2,435)(  3,433)(  4,440)(  5,441)(  6,439)(  7,437)
(  8,438)(  9,436)( 10,452)( 11,453)( 12,451)( 13,458)( 14,459)( 15,457)
( 16,455)( 17,456)( 18,454)( 19,443)( 20,444)( 21,442)( 22,449)( 23,450)
( 24,448)( 25,446)( 26,447)( 27,445)( 28,408)( 29,406)( 30,407)( 31,414)
( 32,412)( 33,413)( 34,411)( 35,409)( 36,410)( 37,426)( 38,424)( 39,425)
( 40,432)( 41,430)( 42,431)( 43,429)( 44,427)( 45,428)( 46,417)( 47,415)
( 48,416)( 49,423)( 50,421)( 51,422)( 52,420)( 53,418)( 54,419)( 55,463)
( 56,464)( 57,465)( 58,460)( 59,461)( 60,462)( 61,466)( 62,467)( 63,468)
( 64,481)( 65,482)( 66,483)( 67,478)( 68,479)( 69,480)( 70,484)( 71,485)
( 72,486)( 73,472)( 74,473)( 75,474)( 76,469)( 77,470)( 78,471)( 79,475)
( 80,476)( 81,477)( 82,353)( 83,354)( 84,352)( 85,359)( 86,360)( 87,358)
( 88,356)( 89,357)( 90,355)( 91,371)( 92,372)( 93,370)( 94,377)( 95,378)
( 96,376)( 97,374)( 98,375)( 99,373)(100,362)(101,363)(102,361)(103,368)
(104,369)(105,367)(106,365)(107,366)(108,364)(109,327)(110,325)(111,326)
(112,333)(113,331)(114,332)(115,330)(116,328)(117,329)(118,345)(119,343)
(120,344)(121,351)(122,349)(123,350)(124,348)(125,346)(126,347)(127,336)
(128,334)(129,335)(130,342)(131,340)(132,341)(133,339)(134,337)(135,338)
(136,382)(137,383)(138,384)(139,379)(140,380)(141,381)(142,385)(143,386)
(144,387)(145,400)(146,401)(147,402)(148,397)(149,398)(150,399)(151,403)
(152,404)(153,405)(154,391)(155,392)(156,393)(157,388)(158,389)(159,390)
(160,394)(161,395)(162,396)(163,272)(164,273)(165,271)(166,278)(167,279)
(168,277)(169,275)(170,276)(171,274)(172,290)(173,291)(174,289)(175,296)
(176,297)(177,295)(178,293)(179,294)(180,292)(181,281)(182,282)(183,280)
(184,287)(185,288)(186,286)(187,284)(188,285)(189,283)(190,246)(191,244)
(192,245)(193,252)(194,250)(195,251)(196,249)(197,247)(198,248)(199,264)
(200,262)(201,263)(202,270)(203,268)(204,269)(205,267)(206,265)(207,266)
(208,255)(209,253)(210,254)(211,261)(212,259)(213,260)(214,258)(215,256)
(216,257)(217,301)(218,302)(219,303)(220,298)(221,299)(222,300)(223,304)
(224,305)(225,306)(226,319)(227,320)(228,321)(229,316)(230,317)(231,318)
(232,322)(233,323)(234,324)(235,310)(236,311)(237,312)(238,307)(239,308)
(240,309)(241,313)(242,314)(243,315);
s2 := 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)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 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)
(109,199)(110,201)(111,200)(112,202)(113,204)(114,203)(115,205)(116,207)
(117,206)(118,190)(119,192)(120,191)(121,193)(122,195)(123,194)(124,196)
(125,198)(126,197)(127,208)(128,210)(129,209)(130,211)(131,213)(132,212)
(133,214)(134,216)(135,215)(136,226)(137,228)(138,227)(139,229)(140,231)
(141,230)(142,232)(143,234)(144,233)(145,217)(146,219)(147,218)(148,220)
(149,222)(150,221)(151,223)(152,225)(153,224)(154,235)(155,237)(156,236)
(157,238)(158,240)(159,239)(160,241)(161,243)(162,242)(245,246)(248,249)
(251,252)(253,262)(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)
(260,270)(261,269)(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)
(283,292)(284,294)(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)
(305,306)(307,316)(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)
(314,324)(315,323)(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)(352,442)(353,444)(354,443)
(355,445)(356,447)(357,446)(358,448)(359,450)(360,449)(361,433)(362,435)
(363,434)(364,436)(365,438)(366,437)(367,439)(368,441)(369,440)(370,451)
(371,453)(372,452)(373,454)(374,456)(375,455)(376,457)(377,459)(378,458)
(379,469)(380,471)(381,470)(382,472)(383,474)(384,473)(385,475)(386,477)
(387,476)(388,460)(389,462)(390,461)(391,463)(392,465)(393,464)(394,466)
(395,468)(396,467)(397,478)(398,480)(399,479)(400,481)(401,483)(402,482)
(403,484)(404,486)(405,485);
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*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, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope