Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944p
if this polytope has a name.
Group : SmallGroup(1944,949)
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}*972f
   3-fold quotients : {6,18}*648c
   6-fold quotients : {6,9}*324c
   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)( 28,144)( 29,143)( 30,142)( 31,137)( 32,136)
( 33,138)( 34,139)( 35,141)( 36,140)( 37,162)( 38,161)( 39,160)( 40,155)
( 41,154)( 42,156)( 43,157)( 44,159)( 45,158)( 46,153)( 47,152)( 48,151)
( 49,146)( 50,145)( 51,147)( 52,148)( 53,150)( 54,149)( 55,113)( 56,112)
( 57,114)( 58,115)( 59,117)( 60,116)( 61,111)( 62,110)( 63,109)( 64,131)
( 65,130)( 66,132)( 67,133)( 68,135)( 69,134)( 70,129)( 71,128)( 72,127)
( 73,122)( 74,121)( 75,123)( 76,124)( 77,126)( 78,125)( 79,120)( 80,119)
( 81,118)(163,172)(164,174)(165,173)(166,177)(167,176)(168,175)(169,179)
(170,178)(171,180)(182,183)(184,186)(187,188)(190,234)(191,233)(192,232)
(193,227)(194,226)(195,228)(196,229)(197,231)(198,230)(199,225)(200,224)
(201,223)(202,218)(203,217)(204,219)(205,220)(206,222)(207,221)(208,243)
(209,242)(210,241)(211,236)(212,235)(213,237)(214,238)(215,240)(216,239)
(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)(271,387)(272,386)(273,385)(274,380)(275,379)
(276,381)(277,382)(278,384)(279,383)(280,405)(281,404)(282,403)(283,398)
(284,397)(285,399)(286,400)(287,402)(288,401)(289,396)(290,395)(291,394)
(292,389)(293,388)(294,390)(295,391)(296,393)(297,392)(298,356)(299,355)
(300,357)(301,358)(302,360)(303,359)(304,354)(305,353)(306,352)(307,374)
(308,373)(309,375)(310,376)(311,378)(312,377)(313,372)(314,371)(315,370)
(316,365)(317,364)(318,366)(319,367)(320,369)(321,368)(322,363)(323,362)
(324,361)(406,415)(407,417)(408,416)(409,420)(410,419)(411,418)(412,422)
(413,421)(414,423)(425,426)(427,429)(430,431)(433,477)(434,476)(435,475)
(436,470)(437,469)(438,471)(439,472)(440,474)(441,473)(442,468)(443,467)
(444,466)(445,461)(446,460)(447,462)(448,463)(449,465)(450,464)(451,486)
(452,485)(453,484)(454,479)(455,478)(456,480)(457,481)(458,483)(459,482);;
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)( 55,298)( 56,300)
( 57,299)( 58,301)( 59,303)( 60,302)( 61,304)( 62,306)( 63,305)( 64,316)
( 65,318)( 66,317)( 67,319)( 68,321)( 69,320)( 70,322)( 71,324)( 72,323)
( 73,307)( 74,309)( 75,308)( 76,310)( 77,312)( 78,311)( 79,313)( 80,315)
( 81,314)( 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)(136,469)
(137,471)(138,470)(139,472)(140,474)(141,473)(142,475)(143,477)(144,476)
(145,460)(146,462)(147,461)(148,463)(149,465)(150,464)(151,466)(152,468)
(153,467)(154,478)(155,480)(156,479)(157,481)(158,483)(159,482)(160,484)
(161,486)(162,485)(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)
(217,388)(218,390)(219,389)(220,391)(221,393)(222,392)(223,394)(224,396)
(225,395)(226,379)(227,381)(228,380)(229,382)(230,384)(231,383)(232,385)
(233,387)(234,386)(235,397)(236,399)(237,398)(238,400)(239,402)(240,401)
(241,403)(242,405)(243,404);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 ];;
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)( 28,144)( 29,143)( 30,142)( 31,137)
( 32,136)( 33,138)( 34,139)( 35,141)( 36,140)( 37,162)( 38,161)( 39,160)
( 40,155)( 41,154)( 42,156)( 43,157)( 44,159)( 45,158)( 46,153)( 47,152)
( 48,151)( 49,146)( 50,145)( 51,147)( 52,148)( 53,150)( 54,149)( 55,113)
( 56,112)( 57,114)( 58,115)( 59,117)( 60,116)( 61,111)( 62,110)( 63,109)
( 64,131)( 65,130)( 66,132)( 67,133)( 68,135)( 69,134)( 70,129)( 71,128)
( 72,127)( 73,122)( 74,121)( 75,123)( 76,124)( 77,126)( 78,125)( 79,120)
( 80,119)( 81,118)(163,172)(164,174)(165,173)(166,177)(167,176)(168,175)
(169,179)(170,178)(171,180)(182,183)(184,186)(187,188)(190,234)(191,233)
(192,232)(193,227)(194,226)(195,228)(196,229)(197,231)(198,230)(199,225)
(200,224)(201,223)(202,218)(203,217)(204,219)(205,220)(206,222)(207,221)
(208,243)(209,242)(210,241)(211,236)(212,235)(213,237)(214,238)(215,240)
(216,239)(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)(271,387)(272,386)(273,385)(274,380)
(275,379)(276,381)(277,382)(278,384)(279,383)(280,405)(281,404)(282,403)
(283,398)(284,397)(285,399)(286,400)(287,402)(288,401)(289,396)(290,395)
(291,394)(292,389)(293,388)(294,390)(295,391)(296,393)(297,392)(298,356)
(299,355)(300,357)(301,358)(302,360)(303,359)(304,354)(305,353)(306,352)
(307,374)(308,373)(309,375)(310,376)(311,378)(312,377)(313,372)(314,371)
(315,370)(316,365)(317,364)(318,366)(319,367)(320,369)(321,368)(322,363)
(323,362)(324,361)(406,415)(407,417)(408,416)(409,420)(410,419)(411,418)
(412,422)(413,421)(414,423)(425,426)(427,429)(430,431)(433,477)(434,476)
(435,475)(436,470)(437,469)(438,471)(439,472)(440,474)(441,473)(442,468)
(443,467)(444,466)(445,461)(446,460)(447,462)(448,463)(449,465)(450,464)
(451,486)(452,485)(453,484)(454,479)(455,478)(456,480)(457,481)(458,483)
(459,482);
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)( 55,298)
( 56,300)( 57,299)( 58,301)( 59,303)( 60,302)( 61,304)( 62,306)( 63,305)
( 64,316)( 65,318)( 66,317)( 67,319)( 68,321)( 69,320)( 70,322)( 71,324)
( 72,323)( 73,307)( 74,309)( 75,308)( 76,310)( 77,312)( 78,311)( 79,313)
( 80,315)( 81,314)( 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)
(136,469)(137,471)(138,470)(139,472)(140,474)(141,473)(142,475)(143,477)
(144,476)(145,460)(146,462)(147,461)(148,463)(149,465)(150,464)(151,466)
(152,468)(153,467)(154,478)(155,480)(156,479)(157,481)(158,483)(159,482)
(160,484)(161,486)(162,485)(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)(217,388)(218,390)(219,389)(220,391)(221,393)(222,392)(223,394)
(224,396)(225,395)(226,379)(227,381)(228,380)(229,382)(230,384)(231,383)
(232,385)(233,387)(234,386)(235,397)(236,399)(237,398)(238,400)(239,402)
(240,401)(241,403)(242,405)(243,404);
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 >;

References : None.
to this polytope