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Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944ab
if this polytope has a name.
Group : SmallGroup(1944,952)
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}*972j
   3-fold quotients : {18,6}*648d, {6,18}*648h
   6-fold quotients : {9,6}*324b, {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)(  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)(  4,441)(  5,440)(  6,439)(  7,434)(  8,433)
(  9,435)( 10,454)( 11,456)( 12,455)( 13,459)( 14,458)( 15,457)( 16,452)
( 17,451)( 18,453)( 19,445)( 20,447)( 21,446)( 22,450)( 23,449)( 24,448)
( 25,443)( 26,442)( 27,444)( 28,413)( 29,412)( 30,414)( 31,406)( 32,408)
( 33,407)( 34,411)( 35,410)( 36,409)( 37,431)( 38,430)( 39,432)( 40,424)
( 41,426)( 42,425)( 43,429)( 44,428)( 45,427)( 46,422)( 47,421)( 48,423)
( 49,415)( 50,417)( 51,416)( 52,420)( 53,419)( 54,418)( 55,460)( 56,462)
( 57,461)( 58,465)( 59,464)( 60,463)( 61,467)( 62,466)( 63,468)( 64,478)
( 65,480)( 66,479)( 67,483)( 68,482)( 69,481)( 70,485)( 71,484)( 72,486)
( 73,469)( 74,471)( 75,470)( 76,474)( 77,473)( 78,472)( 79,476)( 80,475)
( 81,477)( 82,355)( 83,357)( 84,356)( 85,360)( 86,359)( 87,358)( 88,353)
( 89,352)( 90,354)( 91,373)( 92,375)( 93,374)( 94,378)( 95,377)( 96,376)
( 97,371)( 98,370)( 99,372)(100,364)(101,366)(102,365)(103,369)(104,368)
(105,367)(106,362)(107,361)(108,363)(109,332)(110,331)(111,333)(112,325)
(113,327)(114,326)(115,330)(116,329)(117,328)(118,350)(119,349)(120,351)
(121,343)(122,345)(123,344)(124,348)(125,347)(126,346)(127,341)(128,340)
(129,342)(130,334)(131,336)(132,335)(133,339)(134,338)(135,337)(136,379)
(137,381)(138,380)(139,384)(140,383)(141,382)(142,386)(143,385)(144,387)
(145,397)(146,399)(147,398)(148,402)(149,401)(150,400)(151,404)(152,403)
(153,405)(154,388)(155,390)(156,389)(157,393)(158,392)(159,391)(160,395)
(161,394)(162,396)(163,274)(164,276)(165,275)(166,279)(167,278)(168,277)
(169,272)(170,271)(171,273)(172,292)(173,294)(174,293)(175,297)(176,296)
(177,295)(178,290)(179,289)(180,291)(181,283)(182,285)(183,284)(184,288)
(185,287)(186,286)(187,281)(188,280)(189,282)(190,251)(191,250)(192,252)
(193,244)(194,246)(195,245)(196,249)(197,248)(198,247)(199,269)(200,268)
(201,270)(202,262)(203,264)(204,263)(205,267)(206,266)(207,265)(208,260)
(209,259)(210,261)(211,253)(212,255)(213,254)(214,258)(215,257)(216,256)
(217,298)(218,300)(219,299)(220,303)(221,302)(222,301)(223,305)(224,304)
(225,306)(226,316)(227,318)(228,317)(229,321)(230,320)(231,319)(232,323)
(233,322)(234,324)(235,307)(236,309)(237,308)(238,312)(239,311)(240,310)
(241,314)(242,313)(243,315);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)
( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)( 65, 75)
( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 82,172)
( 83,174)( 84,173)( 85,178)( 86,180)( 87,179)( 88,175)( 89,177)( 90,176)
( 91,163)( 92,165)( 93,164)( 94,169)( 95,171)( 96,170)( 97,166)( 98,168)
( 99,167)(100,181)(101,183)(102,182)(103,187)(104,189)(105,188)(106,184)
(107,186)(108,185)(109,199)(110,201)(111,200)(112,205)(113,207)(114,206)
(115,202)(116,204)(117,203)(118,190)(119,192)(120,191)(121,196)(122,198)
(123,197)(124,193)(125,195)(126,194)(127,208)(128,210)(129,209)(130,214)
(131,216)(132,215)(133,211)(134,213)(135,212)(136,226)(137,228)(138,227)
(139,232)(140,234)(141,233)(142,229)(143,231)(144,230)(145,217)(146,219)
(147,218)(148,223)(149,225)(150,224)(151,220)(152,222)(153,221)(154,235)
(155,237)(156,236)(157,241)(158,243)(159,242)(160,238)(161,240)(162,239)
(245,246)(247,250)(248,252)(249,251)(253,262)(254,264)(255,263)(256,268)
(257,270)(258,269)(259,265)(260,267)(261,266)(272,273)(274,277)(275,279)
(276,278)(280,289)(281,291)(282,290)(283,295)(284,297)(285,296)(286,292)
(287,294)(288,293)(299,300)(301,304)(302,306)(303,305)(307,316)(308,318)
(309,317)(310,322)(311,324)(312,323)(313,319)(314,321)(315,320)(325,415)
(326,417)(327,416)(328,421)(329,423)(330,422)(331,418)(332,420)(333,419)
(334,406)(335,408)(336,407)(337,412)(338,414)(339,413)(340,409)(341,411)
(342,410)(343,424)(344,426)(345,425)(346,430)(347,432)(348,431)(349,427)
(350,429)(351,428)(352,442)(353,444)(354,443)(355,448)(356,450)(357,449)
(358,445)(359,447)(360,446)(361,433)(362,435)(363,434)(364,439)(365,441)
(366,440)(367,436)(368,438)(369,437)(370,451)(371,453)(372,452)(373,457)
(374,459)(375,458)(376,454)(377,456)(378,455)(379,469)(380,471)(381,470)
(382,475)(383,477)(384,476)(385,472)(386,474)(387,473)(388,460)(389,462)
(390,461)(391,466)(392,468)(393,467)(394,463)(395,465)(396,464)(397,478)
(398,480)(399,479)(400,484)(401,486)(402,485)(403,481)(404,483)(405,482);;
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, s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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, 
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)(  4,441)(  5,440)(  6,439)(  7,434)
(  8,433)(  9,435)( 10,454)( 11,456)( 12,455)( 13,459)( 14,458)( 15,457)
( 16,452)( 17,451)( 18,453)( 19,445)( 20,447)( 21,446)( 22,450)( 23,449)
( 24,448)( 25,443)( 26,442)( 27,444)( 28,413)( 29,412)( 30,414)( 31,406)
( 32,408)( 33,407)( 34,411)( 35,410)( 36,409)( 37,431)( 38,430)( 39,432)
( 40,424)( 41,426)( 42,425)( 43,429)( 44,428)( 45,427)( 46,422)( 47,421)
( 48,423)( 49,415)( 50,417)( 51,416)( 52,420)( 53,419)( 54,418)( 55,460)
( 56,462)( 57,461)( 58,465)( 59,464)( 60,463)( 61,467)( 62,466)( 63,468)
( 64,478)( 65,480)( 66,479)( 67,483)( 68,482)( 69,481)( 70,485)( 71,484)
( 72,486)( 73,469)( 74,471)( 75,470)( 76,474)( 77,473)( 78,472)( 79,476)
( 80,475)( 81,477)( 82,355)( 83,357)( 84,356)( 85,360)( 86,359)( 87,358)
( 88,353)( 89,352)( 90,354)( 91,373)( 92,375)( 93,374)( 94,378)( 95,377)
( 96,376)( 97,371)( 98,370)( 99,372)(100,364)(101,366)(102,365)(103,369)
(104,368)(105,367)(106,362)(107,361)(108,363)(109,332)(110,331)(111,333)
(112,325)(113,327)(114,326)(115,330)(116,329)(117,328)(118,350)(119,349)
(120,351)(121,343)(122,345)(123,344)(124,348)(125,347)(126,346)(127,341)
(128,340)(129,342)(130,334)(131,336)(132,335)(133,339)(134,338)(135,337)
(136,379)(137,381)(138,380)(139,384)(140,383)(141,382)(142,386)(143,385)
(144,387)(145,397)(146,399)(147,398)(148,402)(149,401)(150,400)(151,404)
(152,403)(153,405)(154,388)(155,390)(156,389)(157,393)(158,392)(159,391)
(160,395)(161,394)(162,396)(163,274)(164,276)(165,275)(166,279)(167,278)
(168,277)(169,272)(170,271)(171,273)(172,292)(173,294)(174,293)(175,297)
(176,296)(177,295)(178,290)(179,289)(180,291)(181,283)(182,285)(183,284)
(184,288)(185,287)(186,286)(187,281)(188,280)(189,282)(190,251)(191,250)
(192,252)(193,244)(194,246)(195,245)(196,249)(197,248)(198,247)(199,269)
(200,268)(201,270)(202,262)(203,264)(204,263)(205,267)(206,266)(207,265)
(208,260)(209,259)(210,261)(211,253)(212,255)(213,254)(214,258)(215,257)
(216,256)(217,298)(218,300)(219,299)(220,303)(221,302)(222,301)(223,305)
(224,304)(225,306)(226,316)(227,318)(228,317)(229,321)(230,320)(231,319)
(232,323)(233,322)(234,324)(235,307)(236,309)(237,308)(238,312)(239,311)
(240,310)(241,314)(242,313)(243,315);
s2 := Sym(486)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 19)( 11, 21)( 12, 20)
( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)
( 43, 49)( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)
( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)
( 82,172)( 83,174)( 84,173)( 85,178)( 86,180)( 87,179)( 88,175)( 89,177)
( 90,176)( 91,163)( 92,165)( 93,164)( 94,169)( 95,171)( 96,170)( 97,166)
( 98,168)( 99,167)(100,181)(101,183)(102,182)(103,187)(104,189)(105,188)
(106,184)(107,186)(108,185)(109,199)(110,201)(111,200)(112,205)(113,207)
(114,206)(115,202)(116,204)(117,203)(118,190)(119,192)(120,191)(121,196)
(122,198)(123,197)(124,193)(125,195)(126,194)(127,208)(128,210)(129,209)
(130,214)(131,216)(132,215)(133,211)(134,213)(135,212)(136,226)(137,228)
(138,227)(139,232)(140,234)(141,233)(142,229)(143,231)(144,230)(145,217)
(146,219)(147,218)(148,223)(149,225)(150,224)(151,220)(152,222)(153,221)
(154,235)(155,237)(156,236)(157,241)(158,243)(159,242)(160,238)(161,240)
(162,239)(245,246)(247,250)(248,252)(249,251)(253,262)(254,264)(255,263)
(256,268)(257,270)(258,269)(259,265)(260,267)(261,266)(272,273)(274,277)
(275,279)(276,278)(280,289)(281,291)(282,290)(283,295)(284,297)(285,296)
(286,292)(287,294)(288,293)(299,300)(301,304)(302,306)(303,305)(307,316)
(308,318)(309,317)(310,322)(311,324)(312,323)(313,319)(314,321)(315,320)
(325,415)(326,417)(327,416)(328,421)(329,423)(330,422)(331,418)(332,420)
(333,419)(334,406)(335,408)(336,407)(337,412)(338,414)(339,413)(340,409)
(341,411)(342,410)(343,424)(344,426)(345,425)(346,430)(347,432)(348,431)
(349,427)(350,429)(351,428)(352,442)(353,444)(354,443)(355,448)(356,450)
(357,449)(358,445)(359,447)(360,446)(361,433)(362,435)(363,434)(364,439)
(365,441)(366,440)(367,436)(368,438)(369,437)(370,451)(371,453)(372,452)
(373,457)(374,459)(375,458)(376,454)(377,456)(378,455)(379,469)(380,471)
(381,470)(382,475)(383,477)(384,476)(385,472)(386,474)(387,473)(388,460)
(389,462)(390,461)(391,466)(392,468)(393,467)(394,463)(395,465)(396,464)
(397,478)(398,480)(399,479)(400,484)(401,486)(402,485)(403,481)(404,483)
(405,482);
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, s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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, 
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