Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944o
if this polytope has a name.
Group : SmallGroup(1944,946)
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}*972e
3-fold quotients : {18,6}*648e
6-fold quotients : {9,6}*324d
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, 56)( 29, 55)( 30, 57)( 31, 62)
( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)( 39, 66)
( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)( 47, 73)
( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)( 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)(109,222)(110,221)(111,220)(112,219)(113,218)(114,217)
(115,225)(116,224)(117,223)(118,231)(119,230)(120,229)(121,228)(122,227)
(123,226)(124,234)(125,233)(126,232)(127,240)(128,239)(129,238)(130,237)
(131,236)(132,235)(133,243)(134,242)(135,241)(136,195)(137,194)(138,193)
(139,192)(140,191)(141,190)(142,198)(143,197)(144,196)(145,204)(146,203)
(147,202)(148,201)(149,200)(150,199)(151,207)(152,206)(153,205)(154,213)
(155,212)(156,211)(157,210)(158,209)(159,208)(160,216)(161,215)(162,214)
(245,246)(247,250)(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)
(263,264)(265,268)(266,270)(267,269)(271,299)(272,298)(273,300)(274,305)
(275,304)(276,306)(277,302)(278,301)(279,303)(280,308)(281,307)(282,309)
(283,314)(284,313)(285,315)(286,311)(287,310)(288,312)(289,317)(290,316)
(291,318)(292,323)(293,322)(294,324)(295,320)(296,319)(297,321)(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)(352,465)(353,464)(354,463)(355,462)(356,461)(357,460)
(358,468)(359,467)(360,466)(361,474)(362,473)(363,472)(364,471)(365,470)
(366,469)(367,477)(368,476)(369,475)(370,483)(371,482)(372,481)(373,480)
(374,479)(375,478)(376,486)(377,485)(378,484)(379,438)(380,437)(381,436)
(382,435)(383,434)(384,433)(385,441)(386,440)(387,439)(388,447)(389,446)
(390,445)(391,444)(392,443)(393,442)(394,450)(395,449)(396,448)(397,456)
(398,455)(399,454)(400,453)(401,452)(402,451)(403,459)(404,458)(405,457);;
s1 := ( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)( 8,447)
( 9,446)( 10,456)( 11,455)( 12,454)( 13,453)( 14,452)( 15,451)( 16,459)
( 17,458)( 18,457)( 19,440)( 20,439)( 21,441)( 22,437)( 23,436)( 24,438)
( 25,434)( 26,433)( 27,435)( 28,431)( 29,430)( 30,432)( 31,428)( 32,427)
( 33,429)( 34,425)( 35,424)( 36,426)( 37,406)( 38,408)( 39,407)( 40,412)
( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,420)( 47,419)( 48,418)
( 49,417)( 50,416)( 51,415)( 52,423)( 53,422)( 54,421)( 55,462)( 56,461)
( 57,460)( 58,468)( 59,467)( 60,466)( 61,465)( 62,464)( 63,463)( 64,473)
( 65,472)( 66,474)( 67,470)( 68,469)( 69,471)( 70,476)( 71,475)( 72,477)
( 73,484)( 74,486)( 75,485)( 76,481)( 77,483)( 78,482)( 79,478)( 80,480)
( 81,479)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)( 88,364)
( 89,366)( 90,365)( 91,375)( 92,374)( 93,373)( 94,372)( 95,371)( 96,370)
( 97,378)( 98,377)( 99,376)(100,359)(101,358)(102,360)(103,356)(104,355)
(105,357)(106,353)(107,352)(108,354)(109,350)(110,349)(111,351)(112,347)
(113,346)(114,348)(115,344)(116,343)(117,345)(118,325)(119,327)(120,326)
(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,339)(128,338)
(129,337)(130,336)(131,335)(132,334)(133,342)(134,341)(135,340)(136,381)
(137,380)(138,379)(139,387)(140,386)(141,385)(142,384)(143,383)(144,382)
(145,392)(146,391)(147,393)(148,389)(149,388)(150,390)(151,395)(152,394)
(153,396)(154,403)(155,405)(156,404)(157,400)(158,402)(159,401)(160,397)
(161,399)(162,398)(163,280)(164,282)(165,281)(166,286)(167,288)(168,287)
(169,283)(170,285)(171,284)(172,294)(173,293)(174,292)(175,291)(176,290)
(177,289)(178,297)(179,296)(180,295)(181,278)(182,277)(183,279)(184,275)
(185,274)(186,276)(187,272)(188,271)(189,273)(190,269)(191,268)(192,270)
(193,266)(194,265)(195,267)(196,263)(197,262)(198,264)(199,244)(200,246)
(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)(208,258)
(209,257)(210,256)(211,255)(212,254)(213,253)(214,261)(215,260)(216,259)
(217,300)(218,299)(219,298)(220,306)(221,305)(222,304)(223,303)(224,302)
(225,301)(226,311)(227,310)(228,312)(229,308)(230,307)(231,309)(232,314)
(233,313)(234,315)(235,322)(236,324)(237,323)(238,319)(239,321)(240,320)
(241,316)(242,318)(243,317);;
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, s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*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, 56)( 29, 55)( 30, 57)
( 31, 62)( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)
( 39, 66)( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)
( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)
( 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)(109,222)(110,221)(111,220)(112,219)(113,218)
(114,217)(115,225)(116,224)(117,223)(118,231)(119,230)(120,229)(121,228)
(122,227)(123,226)(124,234)(125,233)(126,232)(127,240)(128,239)(129,238)
(130,237)(131,236)(132,235)(133,243)(134,242)(135,241)(136,195)(137,194)
(138,193)(139,192)(140,191)(141,190)(142,198)(143,197)(144,196)(145,204)
(146,203)(147,202)(148,201)(149,200)(150,199)(151,207)(152,206)(153,205)
(154,213)(155,212)(156,211)(157,210)(158,209)(159,208)(160,216)(161,215)
(162,214)(245,246)(247,250)(248,252)(249,251)(254,255)(256,259)(257,261)
(258,260)(263,264)(265,268)(266,270)(267,269)(271,299)(272,298)(273,300)
(274,305)(275,304)(276,306)(277,302)(278,301)(279,303)(280,308)(281,307)
(282,309)(283,314)(284,313)(285,315)(286,311)(287,310)(288,312)(289,317)
(290,316)(291,318)(292,323)(293,322)(294,324)(295,320)(296,319)(297,321)
(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)(352,465)(353,464)(354,463)(355,462)(356,461)
(357,460)(358,468)(359,467)(360,466)(361,474)(362,473)(363,472)(364,471)
(365,470)(366,469)(367,477)(368,476)(369,475)(370,483)(371,482)(372,481)
(373,480)(374,479)(375,478)(376,486)(377,485)(378,484)(379,438)(380,437)
(381,436)(382,435)(383,434)(384,433)(385,441)(386,440)(387,439)(388,447)
(389,446)(390,445)(391,444)(392,443)(393,442)(394,450)(395,449)(396,448)
(397,456)(398,455)(399,454)(400,453)(401,452)(402,451)(403,459)(404,458)
(405,457);
s1 := Sym(486)!( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)
( 8,447)( 9,446)( 10,456)( 11,455)( 12,454)( 13,453)( 14,452)( 15,451)
( 16,459)( 17,458)( 18,457)( 19,440)( 20,439)( 21,441)( 22,437)( 23,436)
( 24,438)( 25,434)( 26,433)( 27,435)( 28,431)( 29,430)( 30,432)( 31,428)
( 32,427)( 33,429)( 34,425)( 35,424)( 36,426)( 37,406)( 38,408)( 39,407)
( 40,412)( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,420)( 47,419)
( 48,418)( 49,417)( 50,416)( 51,415)( 52,423)( 53,422)( 54,421)( 55,462)
( 56,461)( 57,460)( 58,468)( 59,467)( 60,466)( 61,465)( 62,464)( 63,463)
( 64,473)( 65,472)( 66,474)( 67,470)( 68,469)( 69,471)( 70,476)( 71,475)
( 72,477)( 73,484)( 74,486)( 75,485)( 76,481)( 77,483)( 78,482)( 79,478)
( 80,480)( 81,479)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)
( 88,364)( 89,366)( 90,365)( 91,375)( 92,374)( 93,373)( 94,372)( 95,371)
( 96,370)( 97,378)( 98,377)( 99,376)(100,359)(101,358)(102,360)(103,356)
(104,355)(105,357)(106,353)(107,352)(108,354)(109,350)(110,349)(111,351)
(112,347)(113,346)(114,348)(115,344)(116,343)(117,345)(118,325)(119,327)
(120,326)(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,339)
(128,338)(129,337)(130,336)(131,335)(132,334)(133,342)(134,341)(135,340)
(136,381)(137,380)(138,379)(139,387)(140,386)(141,385)(142,384)(143,383)
(144,382)(145,392)(146,391)(147,393)(148,389)(149,388)(150,390)(151,395)
(152,394)(153,396)(154,403)(155,405)(156,404)(157,400)(158,402)(159,401)
(160,397)(161,399)(162,398)(163,280)(164,282)(165,281)(166,286)(167,288)
(168,287)(169,283)(170,285)(171,284)(172,294)(173,293)(174,292)(175,291)
(176,290)(177,289)(178,297)(179,296)(180,295)(181,278)(182,277)(183,279)
(184,275)(185,274)(186,276)(187,272)(188,271)(189,273)(190,269)(191,268)
(192,270)(193,266)(194,265)(195,267)(196,263)(197,262)(198,264)(199,244)
(200,246)(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)
(208,258)(209,257)(210,256)(211,255)(212,254)(213,253)(214,261)(215,260)
(216,259)(217,300)(218,299)(219,298)(220,306)(221,305)(222,304)(223,303)
(224,302)(225,301)(226,311)(227,310)(228,312)(229,308)(230,307)(231,309)
(232,314)(233,313)(234,315)(235,322)(236,324)(237,323)(238,319)(239,321)
(240,320)(241,316)(242,318)(243,317);
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, s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1 >;
References : None.
to this polytope