Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944r
if this polytope has a name.
Group : SmallGroup(1944,950)
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 : {18,9}*972g
   3-fold quotients : {6,18}*648e
   6-fold quotients : {6,9}*324d
   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 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)
( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 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)(109,199)
(110,200)(111,201)(112,205)(113,206)(114,207)(115,202)(116,203)(117,204)
(118,190)(119,191)(120,192)(121,196)(122,197)(123,198)(124,193)(125,194)
(126,195)(127,208)(128,209)(129,210)(130,214)(131,215)(132,216)(133,211)
(134,212)(135,213)(136,226)(137,227)(138,228)(139,232)(140,233)(141,234)
(142,229)(143,230)(144,231)(145,217)(146,218)(147,219)(148,223)(149,224)
(150,225)(151,220)(152,221)(153,222)(154,235)(155,236)(156,237)(157,241)
(158,242)(159,243)(160,238)(161,239)(162,240)(247,250)(248,251)(249,252)
(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)(260,266)
(261,267)(274,277)(275,278)(276,279)(280,289)(281,290)(282,291)(283,295)
(284,296)(285,297)(286,292)(287,293)(288,294)(301,304)(302,305)(303,306)
(307,316)(308,317)(309,318)(310,322)(311,323)(312,324)(313,319)(314,320)
(315,321)(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)(352,442)(353,443)(354,444)(355,448)
(356,449)(357,450)(358,445)(359,446)(360,447)(361,433)(362,434)(363,435)
(364,439)(365,440)(366,441)(367,436)(368,437)(369,438)(370,451)(371,452)
(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)(379,469)
(380,470)(381,471)(382,475)(383,476)(384,477)(385,472)(386,473)(387,474)
(388,460)(389,461)(390,462)(391,466)(392,467)(393,468)(394,463)(395,464)
(396,465)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)(403,481)
(404,482)(405,483);;
s1 := (  1, 82)(  2, 84)(  3, 83)(  4, 87)(  5, 86)(  6, 85)(  7, 89)(  8, 88)
(  9, 90)( 10,100)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)( 16,107)
( 17,106)( 18,108)( 19, 91)( 20, 93)( 21, 92)( 22, 96)( 23, 95)( 24, 94)
( 25, 98)( 26, 97)( 27, 99)( 28,142)( 29,144)( 30,143)( 31,138)( 32,137)
( 33,136)( 34,140)( 35,139)( 36,141)( 37,160)( 38,162)( 39,161)( 40,156)
( 41,155)( 42,154)( 43,158)( 44,157)( 45,159)( 46,151)( 47,153)( 48,152)
( 49,147)( 50,146)( 51,145)( 52,149)( 53,148)( 54,150)( 55,114)( 56,113)
( 57,112)( 58,116)( 59,115)( 60,117)( 61,109)( 62,111)( 63,110)( 64,132)
( 65,131)( 66,130)( 67,134)( 68,133)( 69,135)( 70,127)( 71,129)( 72,128)
( 73,123)( 74,122)( 75,121)( 76,125)( 77,124)( 78,126)( 79,118)( 80,120)
( 81,119)(163,172)(164,174)(165,173)(166,177)(167,176)(168,175)(169,179)
(170,178)(171,180)(182,183)(184,186)(187,188)(190,232)(191,234)(192,233)
(193,228)(194,227)(195,226)(196,230)(197,229)(198,231)(199,223)(200,225)
(201,224)(202,219)(203,218)(204,217)(205,221)(206,220)(207,222)(208,241)
(209,243)(210,242)(211,237)(212,236)(213,235)(214,239)(215,238)(216,240)
(244,325)(245,327)(246,326)(247,330)(248,329)(249,328)(250,332)(251,331)
(252,333)(253,343)(254,345)(255,344)(256,348)(257,347)(258,346)(259,350)
(260,349)(261,351)(262,334)(263,336)(264,335)(265,339)(266,338)(267,337)
(268,341)(269,340)(270,342)(271,385)(272,387)(273,386)(274,381)(275,380)
(276,379)(277,383)(278,382)(279,384)(280,403)(281,405)(282,404)(283,399)
(284,398)(285,397)(286,401)(287,400)(288,402)(289,394)(290,396)(291,395)
(292,390)(293,389)(294,388)(295,392)(296,391)(297,393)(298,357)(299,356)
(300,355)(301,359)(302,358)(303,360)(304,352)(305,354)(306,353)(307,375)
(308,374)(309,373)(310,377)(311,376)(312,378)(313,370)(314,372)(315,371)
(316,366)(317,365)(318,364)(319,368)(320,367)(321,369)(322,361)(323,363)
(324,362)(406,415)(407,417)(408,416)(409,420)(410,419)(411,418)(412,422)
(413,421)(414,423)(425,426)(427,429)(430,431)(433,475)(434,477)(435,476)
(436,471)(437,470)(438,469)(439,473)(440,472)(441,474)(442,466)(443,468)
(444,467)(445,462)(446,461)(447,460)(448,464)(449,463)(450,465)(451,484)
(452,486)(453,485)(454,480)(455,479)(456,478)(457,482)(458,481)(459,483);;
s2 := (  1,271)(  2,273)(  3,272)(  4,274)(  5,276)(  6,275)(  7,277)(  8,279)
(  9,278)( 10,289)( 11,291)( 12,290)( 13,292)( 14,294)( 15,293)( 16,295)
( 17,297)( 18,296)( 19,280)( 20,282)( 21,281)( 22,283)( 23,285)( 24,284)
( 25,286)( 26,288)( 27,287)( 28,244)( 29,246)( 30,245)( 31,247)( 32,249)
( 33,248)( 34,250)( 35,252)( 36,251)( 37,262)( 38,264)( 39,263)( 40,265)
( 41,267)( 42,266)( 43,268)( 44,270)( 45,269)( 46,253)( 47,255)( 48,254)
( 49,256)( 50,258)( 51,257)( 52,259)( 53,261)( 54,260)( 55,299)( 56,298)
( 57,300)( 58,302)( 59,301)( 60,303)( 61,305)( 62,304)( 63,306)( 64,317)
( 65,316)( 66,318)( 67,320)( 68,319)( 69,321)( 70,323)( 71,322)( 72,324)
( 73,308)( 74,307)( 75,309)( 76,311)( 77,310)( 78,312)( 79,314)( 80,313)
( 81,315)( 82,442)( 83,444)( 84,443)( 85,445)( 86,447)( 87,446)( 88,448)
( 89,450)( 90,449)( 91,433)( 92,435)( 93,434)( 94,436)( 95,438)( 96,437)
( 97,439)( 98,441)( 99,440)(100,451)(101,453)(102,452)(103,454)(104,456)
(105,455)(106,457)(107,459)(108,458)(109,415)(110,417)(111,416)(112,418)
(113,420)(114,419)(115,421)(116,423)(117,422)(118,406)(119,408)(120,407)
(121,409)(122,411)(123,410)(124,412)(125,414)(126,413)(127,424)(128,426)
(129,425)(130,427)(131,429)(132,428)(133,430)(134,432)(135,431)(136,470)
(137,469)(138,471)(139,473)(140,472)(141,474)(142,476)(143,475)(144,477)
(145,461)(146,460)(147,462)(148,464)(149,463)(150,465)(151,467)(152,466)
(153,468)(154,479)(155,478)(156,480)(157,482)(158,481)(159,483)(160,485)
(161,484)(162,486)(163,361)(164,363)(165,362)(166,364)(167,366)(168,365)
(169,367)(170,369)(171,368)(172,352)(173,354)(174,353)(175,355)(176,357)
(177,356)(178,358)(179,360)(180,359)(181,370)(182,372)(183,371)(184,373)
(185,375)(186,374)(187,376)(188,378)(189,377)(190,334)(191,336)(192,335)
(193,337)(194,339)(195,338)(196,340)(197,342)(198,341)(199,325)(200,327)
(201,326)(202,328)(203,330)(204,329)(205,331)(206,333)(207,332)(208,343)
(209,345)(210,344)(211,346)(212,348)(213,347)(214,349)(215,351)(216,350)
(217,389)(218,388)(219,390)(220,392)(221,391)(222,393)(223,395)(224,394)
(225,396)(226,380)(227,379)(228,381)(229,383)(230,382)(231,384)(232,386)
(233,385)(234,387)(235,398)(236,397)(237,399)(238,401)(239,400)(240,402)
(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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*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 ];;
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)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)
( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 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)
(109,199)(110,200)(111,201)(112,205)(113,206)(114,207)(115,202)(116,203)
(117,204)(118,190)(119,191)(120,192)(121,196)(122,197)(123,198)(124,193)
(125,194)(126,195)(127,208)(128,209)(129,210)(130,214)(131,215)(132,216)
(133,211)(134,212)(135,213)(136,226)(137,227)(138,228)(139,232)(140,233)
(141,234)(142,229)(143,230)(144,231)(145,217)(146,218)(147,219)(148,223)
(149,224)(150,225)(151,220)(152,221)(153,222)(154,235)(155,236)(156,237)
(157,241)(158,242)(159,243)(160,238)(161,239)(162,240)(247,250)(248,251)
(249,252)(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)
(260,266)(261,267)(274,277)(275,278)(276,279)(280,289)(281,290)(282,291)
(283,295)(284,296)(285,297)(286,292)(287,293)(288,294)(301,304)(302,305)
(303,306)(307,316)(308,317)(309,318)(310,322)(311,323)(312,324)(313,319)
(314,320)(315,321)(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)(352,442)(353,443)(354,444)
(355,448)(356,449)(357,450)(358,445)(359,446)(360,447)(361,433)(362,434)
(363,435)(364,439)(365,440)(366,441)(367,436)(368,437)(369,438)(370,451)
(371,452)(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)
(379,469)(380,470)(381,471)(382,475)(383,476)(384,477)(385,472)(386,473)
(387,474)(388,460)(389,461)(390,462)(391,466)(392,467)(393,468)(394,463)
(395,464)(396,465)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)
(403,481)(404,482)(405,483);
s1 := Sym(486)!(  1, 82)(  2, 84)(  3, 83)(  4, 87)(  5, 86)(  6, 85)(  7, 89)
(  8, 88)(  9, 90)( 10,100)( 11,102)( 12,101)( 13,105)( 14,104)( 15,103)
( 16,107)( 17,106)( 18,108)( 19, 91)( 20, 93)( 21, 92)( 22, 96)( 23, 95)
( 24, 94)( 25, 98)( 26, 97)( 27, 99)( 28,142)( 29,144)( 30,143)( 31,138)
( 32,137)( 33,136)( 34,140)( 35,139)( 36,141)( 37,160)( 38,162)( 39,161)
( 40,156)( 41,155)( 42,154)( 43,158)( 44,157)( 45,159)( 46,151)( 47,153)
( 48,152)( 49,147)( 50,146)( 51,145)( 52,149)( 53,148)( 54,150)( 55,114)
( 56,113)( 57,112)( 58,116)( 59,115)( 60,117)( 61,109)( 62,111)( 63,110)
( 64,132)( 65,131)( 66,130)( 67,134)( 68,133)( 69,135)( 70,127)( 71,129)
( 72,128)( 73,123)( 74,122)( 75,121)( 76,125)( 77,124)( 78,126)( 79,118)
( 80,120)( 81,119)(163,172)(164,174)(165,173)(166,177)(167,176)(168,175)
(169,179)(170,178)(171,180)(182,183)(184,186)(187,188)(190,232)(191,234)
(192,233)(193,228)(194,227)(195,226)(196,230)(197,229)(198,231)(199,223)
(200,225)(201,224)(202,219)(203,218)(204,217)(205,221)(206,220)(207,222)
(208,241)(209,243)(210,242)(211,237)(212,236)(213,235)(214,239)(215,238)
(216,240)(244,325)(245,327)(246,326)(247,330)(248,329)(249,328)(250,332)
(251,331)(252,333)(253,343)(254,345)(255,344)(256,348)(257,347)(258,346)
(259,350)(260,349)(261,351)(262,334)(263,336)(264,335)(265,339)(266,338)
(267,337)(268,341)(269,340)(270,342)(271,385)(272,387)(273,386)(274,381)
(275,380)(276,379)(277,383)(278,382)(279,384)(280,403)(281,405)(282,404)
(283,399)(284,398)(285,397)(286,401)(287,400)(288,402)(289,394)(290,396)
(291,395)(292,390)(293,389)(294,388)(295,392)(296,391)(297,393)(298,357)
(299,356)(300,355)(301,359)(302,358)(303,360)(304,352)(305,354)(306,353)
(307,375)(308,374)(309,373)(310,377)(311,376)(312,378)(313,370)(314,372)
(315,371)(316,366)(317,365)(318,364)(319,368)(320,367)(321,369)(322,361)
(323,363)(324,362)(406,415)(407,417)(408,416)(409,420)(410,419)(411,418)
(412,422)(413,421)(414,423)(425,426)(427,429)(430,431)(433,475)(434,477)
(435,476)(436,471)(437,470)(438,469)(439,473)(440,472)(441,474)(442,466)
(443,468)(444,467)(445,462)(446,461)(447,460)(448,464)(449,463)(450,465)
(451,484)(452,486)(453,485)(454,480)(455,479)(456,478)(457,482)(458,481)
(459,483);
s2 := Sym(486)!(  1,271)(  2,273)(  3,272)(  4,274)(  5,276)(  6,275)(  7,277)
(  8,279)(  9,278)( 10,289)( 11,291)( 12,290)( 13,292)( 14,294)( 15,293)
( 16,295)( 17,297)( 18,296)( 19,280)( 20,282)( 21,281)( 22,283)( 23,285)
( 24,284)( 25,286)( 26,288)( 27,287)( 28,244)( 29,246)( 30,245)( 31,247)
( 32,249)( 33,248)( 34,250)( 35,252)( 36,251)( 37,262)( 38,264)( 39,263)
( 40,265)( 41,267)( 42,266)( 43,268)( 44,270)( 45,269)( 46,253)( 47,255)
( 48,254)( 49,256)( 50,258)( 51,257)( 52,259)( 53,261)( 54,260)( 55,299)
( 56,298)( 57,300)( 58,302)( 59,301)( 60,303)( 61,305)( 62,304)( 63,306)
( 64,317)( 65,316)( 66,318)( 67,320)( 68,319)( 69,321)( 70,323)( 71,322)
( 72,324)( 73,308)( 74,307)( 75,309)( 76,311)( 77,310)( 78,312)( 79,314)
( 80,313)( 81,315)( 82,442)( 83,444)( 84,443)( 85,445)( 86,447)( 87,446)
( 88,448)( 89,450)( 90,449)( 91,433)( 92,435)( 93,434)( 94,436)( 95,438)
( 96,437)( 97,439)( 98,441)( 99,440)(100,451)(101,453)(102,452)(103,454)
(104,456)(105,455)(106,457)(107,459)(108,458)(109,415)(110,417)(111,416)
(112,418)(113,420)(114,419)(115,421)(116,423)(117,422)(118,406)(119,408)
(120,407)(121,409)(122,411)(123,410)(124,412)(125,414)(126,413)(127,424)
(128,426)(129,425)(130,427)(131,429)(132,428)(133,430)(134,432)(135,431)
(136,470)(137,469)(138,471)(139,473)(140,472)(141,474)(142,476)(143,475)
(144,477)(145,461)(146,460)(147,462)(148,464)(149,463)(150,465)(151,467)
(152,466)(153,468)(154,479)(155,478)(156,480)(157,482)(158,481)(159,483)
(160,485)(161,484)(162,486)(163,361)(164,363)(165,362)(166,364)(167,366)
(168,365)(169,367)(170,369)(171,368)(172,352)(173,354)(174,353)(175,355)
(176,357)(177,356)(178,358)(179,360)(180,359)(181,370)(182,372)(183,371)
(184,373)(185,375)(186,374)(187,376)(188,378)(189,377)(190,334)(191,336)
(192,335)(193,337)(194,339)(195,338)(196,340)(197,342)(198,341)(199,325)
(200,327)(201,326)(202,328)(203,330)(204,329)(205,331)(206,333)(207,332)
(208,343)(209,345)(210,344)(211,346)(212,348)(213,347)(214,349)(215,351)
(216,350)(217,389)(218,388)(219,390)(220,392)(221,391)(222,393)(223,395)
(224,394)(225,396)(226,380)(227,379)(228,381)(229,383)(230,382)(231,384)
(232,386)(233,385)(234,387)(235,398)(236,397)(237,399)(238,401)(239,400)
(240,402)(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*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*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 >;

References : None.
to this polytope