Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944l
if this polytope has a name.
Group : SmallGroup(1944,945)
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}*972d
3-fold quotients : {18,6}*648e, {6,18}*648h
6-fold quotients : {9,6}*324d, {3,18}*324
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)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 57)( 29, 56)( 30, 55)( 31, 63)
( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)( 39, 64)
( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 75)( 47, 74)
( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 82,167)
( 83,166)( 84,168)( 85,164)( 86,163)( 87,165)( 88,170)( 89,169)( 90,171)
( 91,176)( 92,175)( 93,177)( 94,173)( 95,172)( 96,174)( 97,179)( 98,178)
( 99,180)(100,185)(101,184)(102,186)(103,182)(104,181)(105,183)(106,188)
(107,187)(108,189)(109,220)(110,222)(111,221)(112,217)(113,219)(114,218)
(115,223)(116,225)(117,224)(118,229)(119,231)(120,230)(121,226)(122,228)
(123,227)(124,232)(125,234)(126,233)(127,238)(128,240)(129,239)(130,235)
(131,237)(132,236)(133,241)(134,243)(135,242)(136,193)(137,195)(138,194)
(139,190)(140,192)(141,191)(142,196)(143,198)(144,197)(145,202)(146,204)
(147,203)(148,199)(149,201)(150,200)(151,205)(152,207)(153,206)(154,211)
(155,213)(156,212)(157,208)(158,210)(159,209)(160,214)(161,216)(162,215)
(245,246)(247,250)(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)
(263,264)(265,268)(266,270)(267,269)(271,300)(272,299)(273,298)(274,306)
(275,305)(276,304)(277,303)(278,302)(279,301)(280,309)(281,308)(282,307)
(283,315)(284,314)(285,313)(286,312)(287,311)(288,310)(289,318)(290,317)
(291,316)(292,324)(293,323)(294,322)(295,321)(296,320)(297,319)(325,410)
(326,409)(327,411)(328,407)(329,406)(330,408)(331,413)(332,412)(333,414)
(334,419)(335,418)(336,420)(337,416)(338,415)(339,417)(340,422)(341,421)
(342,423)(343,428)(344,427)(345,429)(346,425)(347,424)(348,426)(349,431)
(350,430)(351,432)(352,463)(353,465)(354,464)(355,460)(356,462)(357,461)
(358,466)(359,468)(360,467)(361,472)(362,474)(363,473)(364,469)(365,471)
(366,470)(367,475)(368,477)(369,476)(370,481)(371,483)(372,482)(373,478)
(374,480)(375,479)(376,484)(377,486)(378,485)(379,436)(380,438)(381,437)
(382,433)(383,435)(384,434)(385,439)(386,441)(387,440)(388,445)(389,447)
(390,446)(391,442)(392,444)(393,443)(394,448)(395,450)(396,449)(397,454)
(398,456)(399,455)(400,451)(401,453)(402,452)(403,457)(404,459)(405,458);;
s1 := ( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)( 8,447)
( 9,446)( 10,458)( 11,457)( 12,459)( 13,455)( 14,454)( 15,456)( 16,452)
( 17,451)( 18,453)( 19,438)( 20,437)( 21,436)( 22,435)( 23,434)( 24,433)
( 25,441)( 26,440)( 27,439)( 28,429)( 29,428)( 30,427)( 31,426)( 32,425)
( 33,424)( 34,432)( 35,431)( 36,430)( 37,406)( 38,408)( 39,407)( 40,412)
( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,422)( 47,421)( 48,423)
( 49,419)( 50,418)( 51,420)( 52,416)( 53,415)( 54,417)( 55,461)( 56,460)
( 57,462)( 58,467)( 59,466)( 60,468)( 61,464)( 62,463)( 63,465)( 64,477)
( 65,476)( 66,475)( 67,474)( 68,473)( 69,472)( 70,471)( 71,470)( 72,469)
( 73,481)( 74,483)( 75,482)( 76,478)( 77,480)( 78,479)( 79,484)( 80,486)
( 81,485)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)( 88,364)
( 89,366)( 90,365)( 91,377)( 92,376)( 93,378)( 94,374)( 95,373)( 96,375)
( 97,371)( 98,370)( 99,372)(100,357)(101,356)(102,355)(103,354)(104,353)
(105,352)(106,360)(107,359)(108,358)(109,348)(110,347)(111,346)(112,345)
(113,344)(114,343)(115,351)(116,350)(117,349)(118,325)(119,327)(120,326)
(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,341)(128,340)
(129,342)(130,338)(131,337)(132,339)(133,335)(134,334)(135,336)(136,380)
(137,379)(138,381)(139,386)(140,385)(141,387)(142,383)(143,382)(144,384)
(145,396)(146,395)(147,394)(148,393)(149,392)(150,391)(151,390)(152,389)
(153,388)(154,400)(155,402)(156,401)(157,397)(158,399)(159,398)(160,403)
(161,405)(162,404)(163,280)(164,282)(165,281)(166,286)(167,288)(168,287)
(169,283)(170,285)(171,284)(172,296)(173,295)(174,297)(175,293)(176,292)
(177,294)(178,290)(179,289)(180,291)(181,276)(182,275)(183,274)(184,273)
(185,272)(186,271)(187,279)(188,278)(189,277)(190,267)(191,266)(192,265)
(193,264)(194,263)(195,262)(196,270)(197,269)(198,268)(199,244)(200,246)
(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)(208,260)
(209,259)(210,261)(211,257)(212,256)(213,258)(214,254)(215,253)(216,255)
(217,299)(218,298)(219,300)(220,305)(221,304)(222,306)(223,302)(224,301)
(225,303)(226,315)(227,314)(228,313)(229,312)(230,311)(231,310)(232,309)
(233,308)(234,307)(235,319)(236,321)(237,320)(238,316)(239,318)(240,317)
(241,322)(242,324)(243,323);;
s2 := ( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)( 85,165)
( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)( 93,184)
( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)(101,177)
(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)(109,194)
(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)(117,198)
(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)(125,215)
(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)(133,205)
(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)(141,218)
(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)(149,235)
(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)(157,228)
(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)(249,250)
(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)(260,265)
(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)(283,296)
(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)(303,304)
(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)(314,319)
(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)(331,412)
(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)(339,425)
(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)(347,415)
(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)(355,435)
(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)(363,454)
(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)(371,447)
(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)(379,464)
(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)(387,468)
(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)(395,485)
(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)(403,475)
(404,476)(405,477);;
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*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s0*s1*s0,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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, 57)( 29, 56)( 30, 55)
( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)
( 39, 64)( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 75)
( 47, 74)( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)
( 82,167)( 83,166)( 84,168)( 85,164)( 86,163)( 87,165)( 88,170)( 89,169)
( 90,171)( 91,176)( 92,175)( 93,177)( 94,173)( 95,172)( 96,174)( 97,179)
( 98,178)( 99,180)(100,185)(101,184)(102,186)(103,182)(104,181)(105,183)
(106,188)(107,187)(108,189)(109,220)(110,222)(111,221)(112,217)(113,219)
(114,218)(115,223)(116,225)(117,224)(118,229)(119,231)(120,230)(121,226)
(122,228)(123,227)(124,232)(125,234)(126,233)(127,238)(128,240)(129,239)
(130,235)(131,237)(132,236)(133,241)(134,243)(135,242)(136,193)(137,195)
(138,194)(139,190)(140,192)(141,191)(142,196)(143,198)(144,197)(145,202)
(146,204)(147,203)(148,199)(149,201)(150,200)(151,205)(152,207)(153,206)
(154,211)(155,213)(156,212)(157,208)(158,210)(159,209)(160,214)(161,216)
(162,215)(245,246)(247,250)(248,252)(249,251)(254,255)(256,259)(257,261)
(258,260)(263,264)(265,268)(266,270)(267,269)(271,300)(272,299)(273,298)
(274,306)(275,305)(276,304)(277,303)(278,302)(279,301)(280,309)(281,308)
(282,307)(283,315)(284,314)(285,313)(286,312)(287,311)(288,310)(289,318)
(290,317)(291,316)(292,324)(293,323)(294,322)(295,321)(296,320)(297,319)
(325,410)(326,409)(327,411)(328,407)(329,406)(330,408)(331,413)(332,412)
(333,414)(334,419)(335,418)(336,420)(337,416)(338,415)(339,417)(340,422)
(341,421)(342,423)(343,428)(344,427)(345,429)(346,425)(347,424)(348,426)
(349,431)(350,430)(351,432)(352,463)(353,465)(354,464)(355,460)(356,462)
(357,461)(358,466)(359,468)(360,467)(361,472)(362,474)(363,473)(364,469)
(365,471)(366,470)(367,475)(368,477)(369,476)(370,481)(371,483)(372,482)
(373,478)(374,480)(375,479)(376,484)(377,486)(378,485)(379,436)(380,438)
(381,437)(382,433)(383,435)(384,434)(385,439)(386,441)(387,440)(388,445)
(389,447)(390,446)(391,442)(392,444)(393,443)(394,448)(395,450)(396,449)
(397,454)(398,456)(399,455)(400,451)(401,453)(402,452)(403,457)(404,459)
(405,458);
s1 := Sym(486)!( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)
( 8,447)( 9,446)( 10,458)( 11,457)( 12,459)( 13,455)( 14,454)( 15,456)
( 16,452)( 17,451)( 18,453)( 19,438)( 20,437)( 21,436)( 22,435)( 23,434)
( 24,433)( 25,441)( 26,440)( 27,439)( 28,429)( 29,428)( 30,427)( 31,426)
( 32,425)( 33,424)( 34,432)( 35,431)( 36,430)( 37,406)( 38,408)( 39,407)
( 40,412)( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,422)( 47,421)
( 48,423)( 49,419)( 50,418)( 51,420)( 52,416)( 53,415)( 54,417)( 55,461)
( 56,460)( 57,462)( 58,467)( 59,466)( 60,468)( 61,464)( 62,463)( 63,465)
( 64,477)( 65,476)( 66,475)( 67,474)( 68,473)( 69,472)( 70,471)( 71,470)
( 72,469)( 73,481)( 74,483)( 75,482)( 76,478)( 77,480)( 78,479)( 79,484)
( 80,486)( 81,485)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)
( 88,364)( 89,366)( 90,365)( 91,377)( 92,376)( 93,378)( 94,374)( 95,373)
( 96,375)( 97,371)( 98,370)( 99,372)(100,357)(101,356)(102,355)(103,354)
(104,353)(105,352)(106,360)(107,359)(108,358)(109,348)(110,347)(111,346)
(112,345)(113,344)(114,343)(115,351)(116,350)(117,349)(118,325)(119,327)
(120,326)(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,341)
(128,340)(129,342)(130,338)(131,337)(132,339)(133,335)(134,334)(135,336)
(136,380)(137,379)(138,381)(139,386)(140,385)(141,387)(142,383)(143,382)
(144,384)(145,396)(146,395)(147,394)(148,393)(149,392)(150,391)(151,390)
(152,389)(153,388)(154,400)(155,402)(156,401)(157,397)(158,399)(159,398)
(160,403)(161,405)(162,404)(163,280)(164,282)(165,281)(166,286)(167,288)
(168,287)(169,283)(170,285)(171,284)(172,296)(173,295)(174,297)(175,293)
(176,292)(177,294)(178,290)(179,289)(180,291)(181,276)(182,275)(183,274)
(184,273)(185,272)(186,271)(187,279)(188,278)(189,277)(190,267)(191,266)
(192,265)(193,264)(194,263)(195,262)(196,270)(197,269)(198,268)(199,244)
(200,246)(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)
(208,260)(209,259)(210,261)(211,257)(212,256)(213,258)(214,254)(215,253)
(216,255)(217,299)(218,298)(219,300)(220,305)(221,304)(222,306)(223,302)
(224,301)(225,303)(226,315)(227,314)(228,313)(229,312)(230,311)(231,310)
(232,309)(233,308)(234,307)(235,319)(236,321)(237,320)(238,316)(239,318)
(240,317)(241,322)(242,324)(243,323);
s2 := Sym(486)!( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)
( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)
( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)
( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)
( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)
( 85,165)( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)
( 93,184)( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)
(101,177)(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)
(109,194)(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)
(117,198)(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)
(125,215)(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)
(133,205)(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)
(141,218)(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)
(149,235)(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)
(157,228)(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)
(249,250)(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)
(260,265)(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)
(283,296)(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)
(303,304)(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)
(314,319)(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)
(331,412)(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)
(339,425)(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)
(347,415)(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)
(355,435)(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)
(363,454)(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)
(371,447)(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)
(379,464)(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)
(387,468)(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)
(395,485)(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)
(403,475)(404,476)(405,477);
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*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s0*s1*s0,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;
References : None.
to this polytope