Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944x
if this polytope has a name.
Group : SmallGroup(1944,951)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
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 : {18,18}*972h
   3-fold quotients : {18,6}*648b, {6,18}*648g
   6-fold quotients : {18,6}*324a, {6,18}*324c
   9-fold quotients : {18,6}*216a, {6,6}*216b
   18-fold quotients : {6,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72a
   54-fold quotients : {9,2}*36
   81-fold quotients : {2,6}*24, {6,2}*24
   162-fold quotients : {2,3}*12, {3,2}*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, 86)(  5, 85)(  6, 87)(  7, 90)(  8, 89)
(  9, 88)( 10,100)( 11,102)( 12,101)( 13,104)( 14,103)( 15,105)( 16,108)
( 17,107)( 18,106)( 19, 91)( 20, 93)( 21, 92)( 22, 95)( 23, 94)( 24, 96)
( 25, 99)( 26, 98)( 27, 97)( 28,144)( 29,143)( 30,142)( 31,136)( 32,138)
( 33,137)( 34,140)( 35,139)( 36,141)( 37,162)( 38,161)( 39,160)( 40,154)
( 41,156)( 42,155)( 43,158)( 44,157)( 45,159)( 46,153)( 47,152)( 48,151)
( 49,145)( 50,147)( 51,146)( 52,149)( 53,148)( 54,150)( 55,112)( 56,114)
( 57,113)( 58,116)( 59,115)( 60,117)( 61,111)( 62,110)( 63,109)( 64,130)
( 65,132)( 66,131)( 67,134)( 68,133)( 69,135)( 70,129)( 71,128)( 72,127)
( 73,121)( 74,123)( 75,122)( 76,125)( 77,124)( 78,126)( 79,120)( 80,119)
( 81,118)(163,172)(164,174)(165,173)(166,176)(167,175)(168,177)(169,180)
(170,179)(171,178)(182,183)(184,185)(187,189)(190,234)(191,233)(192,232)
(193,226)(194,228)(195,227)(196,230)(197,229)(198,231)(199,225)(200,224)
(201,223)(202,217)(203,219)(204,218)(205,221)(206,220)(207,222)(208,243)
(209,242)(210,241)(211,235)(212,237)(213,236)(214,239)(215,238)(216,240)
(244,325)(245,327)(246,326)(247,329)(248,328)(249,330)(250,333)(251,332)
(252,331)(253,343)(254,345)(255,344)(256,347)(257,346)(258,348)(259,351)
(260,350)(261,349)(262,334)(263,336)(264,335)(265,338)(266,337)(267,339)
(268,342)(269,341)(270,340)(271,387)(272,386)(273,385)(274,379)(275,381)
(276,380)(277,383)(278,382)(279,384)(280,405)(281,404)(282,403)(283,397)
(284,399)(285,398)(286,401)(287,400)(288,402)(289,396)(290,395)(291,394)
(292,388)(293,390)(294,389)(295,392)(296,391)(297,393)(298,355)(299,357)
(300,356)(301,359)(302,358)(303,360)(304,354)(305,353)(306,352)(307,373)
(308,375)(309,374)(310,377)(311,376)(312,378)(313,372)(314,371)(315,370)
(316,364)(317,366)(318,365)(319,368)(320,367)(321,369)(322,363)(323,362)
(324,361)(406,415)(407,417)(408,416)(409,419)(410,418)(411,420)(412,423)
(413,422)(414,421)(425,426)(427,428)(430,432)(433,477)(434,476)(435,475)
(436,469)(437,471)(438,470)(439,473)(440,472)(441,474)(442,468)(443,467)
(444,466)(445,460)(446,462)(447,461)(448,464)(449,463)(450,465)(451,486)
(452,485)(453,484)(454,478)(455,480)(456,479)(457,482)(458,481)(459,483);;
s2 := (  1,271)(  2,273)(  3,272)(  4,277)(  5,279)(  6,278)(  7,274)(  8,276)
(  9,275)( 10,280)( 11,282)( 12,281)( 13,286)( 14,288)( 15,287)( 16,283)
( 17,285)( 18,284)( 19,289)( 20,291)( 21,290)( 22,295)( 23,297)( 24,296)
( 25,292)( 26,294)( 27,293)( 28,244)( 29,246)( 30,245)( 31,250)( 32,252)
( 33,251)( 34,247)( 35,249)( 36,248)( 37,253)( 38,255)( 39,254)( 40,259)
( 41,261)( 42,260)( 43,256)( 44,258)( 45,257)( 46,262)( 47,264)( 48,263)
( 49,268)( 50,270)( 51,269)( 52,265)( 53,267)( 54,266)( 55,299)( 56,298)
( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)( 64,308)
( 65,307)( 66,309)( 67,314)( 68,313)( 69,315)( 70,311)( 71,310)( 72,312)
( 73,317)( 74,316)( 75,318)( 76,323)( 77,322)( 78,324)( 79,320)( 80,319)
( 81,321)( 82,352)( 83,354)( 84,353)( 85,358)( 86,360)( 87,359)( 88,355)
( 89,357)( 90,356)( 91,361)( 92,363)( 93,362)( 94,367)( 95,369)( 96,368)
( 97,364)( 98,366)( 99,365)(100,370)(101,372)(102,371)(103,376)(104,378)
(105,377)(106,373)(107,375)(108,374)(109,325)(110,327)(111,326)(112,331)
(113,333)(114,332)(115,328)(116,330)(117,329)(118,334)(119,336)(120,335)
(121,340)(122,342)(123,341)(124,337)(125,339)(126,338)(127,343)(128,345)
(129,344)(130,349)(131,351)(132,350)(133,346)(134,348)(135,347)(136,380)
(137,379)(138,381)(139,386)(140,385)(141,387)(142,383)(143,382)(144,384)
(145,389)(146,388)(147,390)(148,395)(149,394)(150,396)(151,392)(152,391)
(153,393)(154,398)(155,397)(156,399)(157,404)(158,403)(159,405)(160,401)
(161,400)(162,402)(163,433)(164,435)(165,434)(166,439)(167,441)(168,440)
(169,436)(170,438)(171,437)(172,442)(173,444)(174,443)(175,448)(176,450)
(177,449)(178,445)(179,447)(180,446)(181,451)(182,453)(183,452)(184,457)
(185,459)(186,458)(187,454)(188,456)(189,455)(190,406)(191,408)(192,407)
(193,412)(194,414)(195,413)(196,409)(197,411)(198,410)(199,415)(200,417)
(201,416)(202,421)(203,423)(204,422)(205,418)(206,420)(207,419)(208,424)
(209,426)(210,425)(211,430)(212,432)(213,431)(214,427)(215,429)(216,428)
(217,461)(218,460)(219,462)(220,467)(221,466)(222,468)(223,464)(224,463)
(225,465)(226,470)(227,469)(228,471)(229,476)(230,475)(231,477)(232,473)
(233,472)(234,474)(235,479)(236,478)(237,480)(238,485)(239,484)(240,486)
(241,482)(242,481)(243,483);;
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*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*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 ];;
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, 86)(  5, 85)(  6, 87)(  7, 90)
(  8, 89)(  9, 88)( 10,100)( 11,102)( 12,101)( 13,104)( 14,103)( 15,105)
( 16,108)( 17,107)( 18,106)( 19, 91)( 20, 93)( 21, 92)( 22, 95)( 23, 94)
( 24, 96)( 25, 99)( 26, 98)( 27, 97)( 28,144)( 29,143)( 30,142)( 31,136)
( 32,138)( 33,137)( 34,140)( 35,139)( 36,141)( 37,162)( 38,161)( 39,160)
( 40,154)( 41,156)( 42,155)( 43,158)( 44,157)( 45,159)( 46,153)( 47,152)
( 48,151)( 49,145)( 50,147)( 51,146)( 52,149)( 53,148)( 54,150)( 55,112)
( 56,114)( 57,113)( 58,116)( 59,115)( 60,117)( 61,111)( 62,110)( 63,109)
( 64,130)( 65,132)( 66,131)( 67,134)( 68,133)( 69,135)( 70,129)( 71,128)
( 72,127)( 73,121)( 74,123)( 75,122)( 76,125)( 77,124)( 78,126)( 79,120)
( 80,119)( 81,118)(163,172)(164,174)(165,173)(166,176)(167,175)(168,177)
(169,180)(170,179)(171,178)(182,183)(184,185)(187,189)(190,234)(191,233)
(192,232)(193,226)(194,228)(195,227)(196,230)(197,229)(198,231)(199,225)
(200,224)(201,223)(202,217)(203,219)(204,218)(205,221)(206,220)(207,222)
(208,243)(209,242)(210,241)(211,235)(212,237)(213,236)(214,239)(215,238)
(216,240)(244,325)(245,327)(246,326)(247,329)(248,328)(249,330)(250,333)
(251,332)(252,331)(253,343)(254,345)(255,344)(256,347)(257,346)(258,348)
(259,351)(260,350)(261,349)(262,334)(263,336)(264,335)(265,338)(266,337)
(267,339)(268,342)(269,341)(270,340)(271,387)(272,386)(273,385)(274,379)
(275,381)(276,380)(277,383)(278,382)(279,384)(280,405)(281,404)(282,403)
(283,397)(284,399)(285,398)(286,401)(287,400)(288,402)(289,396)(290,395)
(291,394)(292,388)(293,390)(294,389)(295,392)(296,391)(297,393)(298,355)
(299,357)(300,356)(301,359)(302,358)(303,360)(304,354)(305,353)(306,352)
(307,373)(308,375)(309,374)(310,377)(311,376)(312,378)(313,372)(314,371)
(315,370)(316,364)(317,366)(318,365)(319,368)(320,367)(321,369)(322,363)
(323,362)(324,361)(406,415)(407,417)(408,416)(409,419)(410,418)(411,420)
(412,423)(413,422)(414,421)(425,426)(427,428)(430,432)(433,477)(434,476)
(435,475)(436,469)(437,471)(438,470)(439,473)(440,472)(441,474)(442,468)
(443,467)(444,466)(445,460)(446,462)(447,461)(448,464)(449,463)(450,465)
(451,486)(452,485)(453,484)(454,478)(455,480)(456,479)(457,482)(458,481)
(459,483);
s2 := Sym(486)!(  1,271)(  2,273)(  3,272)(  4,277)(  5,279)(  6,278)(  7,274)
(  8,276)(  9,275)( 10,280)( 11,282)( 12,281)( 13,286)( 14,288)( 15,287)
( 16,283)( 17,285)( 18,284)( 19,289)( 20,291)( 21,290)( 22,295)( 23,297)
( 24,296)( 25,292)( 26,294)( 27,293)( 28,244)( 29,246)( 30,245)( 31,250)
( 32,252)( 33,251)( 34,247)( 35,249)( 36,248)( 37,253)( 38,255)( 39,254)
( 40,259)( 41,261)( 42,260)( 43,256)( 44,258)( 45,257)( 46,262)( 47,264)
( 48,263)( 49,268)( 50,270)( 51,269)( 52,265)( 53,267)( 54,266)( 55,299)
( 56,298)( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)
( 64,308)( 65,307)( 66,309)( 67,314)( 68,313)( 69,315)( 70,311)( 71,310)
( 72,312)( 73,317)( 74,316)( 75,318)( 76,323)( 77,322)( 78,324)( 79,320)
( 80,319)( 81,321)( 82,352)( 83,354)( 84,353)( 85,358)( 86,360)( 87,359)
( 88,355)( 89,357)( 90,356)( 91,361)( 92,363)( 93,362)( 94,367)( 95,369)
( 96,368)( 97,364)( 98,366)( 99,365)(100,370)(101,372)(102,371)(103,376)
(104,378)(105,377)(106,373)(107,375)(108,374)(109,325)(110,327)(111,326)
(112,331)(113,333)(114,332)(115,328)(116,330)(117,329)(118,334)(119,336)
(120,335)(121,340)(122,342)(123,341)(124,337)(125,339)(126,338)(127,343)
(128,345)(129,344)(130,349)(131,351)(132,350)(133,346)(134,348)(135,347)
(136,380)(137,379)(138,381)(139,386)(140,385)(141,387)(142,383)(143,382)
(144,384)(145,389)(146,388)(147,390)(148,395)(149,394)(150,396)(151,392)
(152,391)(153,393)(154,398)(155,397)(156,399)(157,404)(158,403)(159,405)
(160,401)(161,400)(162,402)(163,433)(164,435)(165,434)(166,439)(167,441)
(168,440)(169,436)(170,438)(171,437)(172,442)(173,444)(174,443)(175,448)
(176,450)(177,449)(178,445)(179,447)(180,446)(181,451)(182,453)(183,452)
(184,457)(185,459)(186,458)(187,454)(188,456)(189,455)(190,406)(191,408)
(192,407)(193,412)(194,414)(195,413)(196,409)(197,411)(198,410)(199,415)
(200,417)(201,416)(202,421)(203,423)(204,422)(205,418)(206,420)(207,419)
(208,424)(209,426)(210,425)(211,430)(212,432)(213,431)(214,427)(215,429)
(216,428)(217,461)(218,460)(219,462)(220,467)(221,466)(222,468)(223,464)
(224,463)(225,465)(226,470)(227,469)(228,471)(229,476)(230,475)(231,477)
(232,473)(233,472)(234,474)(235,479)(236,478)(237,480)(238,485)(239,484)
(240,486)(241,482)(242,481)(243,483);
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*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*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 >;

References : None.
to this polytope