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