Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944j
if this polytope has a name.
Group : SmallGroup(1944,945)
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
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,9}*972d
3-fold quotients : {6,18}*648e, {18,6}*648h
6-fold quotients : {6,9}*324d, {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 := ( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)( 85,165)
( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)( 93,184)
( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)(101,177)
(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)(109,194)
(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)(117,198)
(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)(125,215)
(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)(133,205)
(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)(141,218)
(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)(149,235)
(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)(157,228)
(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)(249,250)
(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)(260,265)
(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)(283,296)
(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)(303,304)
(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)(314,319)
(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)(331,412)
(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)(339,425)
(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)(347,415)
(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)(355,435)
(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)(363,454)
(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)(371,447)
(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)(379,464)
(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)(387,468)
(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)(395,485)
(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)(403,475)
(404,476)(405,477);;
s1 := ( 1, 82)( 2, 84)( 3, 83)( 4, 88)( 5, 90)( 6, 89)( 7, 85)( 8, 87)
( 9, 86)( 10, 92)( 11, 91)( 12, 93)( 13, 98)( 14, 97)( 15, 99)( 16, 95)
( 17, 94)( 18, 96)( 19,102)( 20,101)( 21,100)( 22,108)( 23,107)( 24,106)
( 25,105)( 26,104)( 27,103)( 28,158)( 29,157)( 30,159)( 31,155)( 32,154)
( 33,156)( 34,161)( 35,160)( 36,162)( 37,141)( 38,140)( 39,139)( 40,138)
( 41,137)( 42,136)( 43,144)( 44,143)( 45,142)( 46,148)( 47,150)( 48,149)
( 49,145)( 50,147)( 51,146)( 52,151)( 53,153)( 54,152)( 55,123)( 56,122)
( 57,121)( 58,120)( 59,119)( 60,118)( 61,126)( 62,125)( 63,124)( 64,130)
( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,133)( 71,135)( 72,134)
( 73,113)( 74,112)( 75,114)( 76,110)( 77,109)( 78,111)( 79,116)( 80,115)
( 81,117)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)(174,175)
(178,180)(181,184)(182,186)(183,185)(188,189)(190,243)(191,242)(192,241)
(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,223)(200,225)
(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)(208,233)
(209,232)(210,234)(211,230)(212,229)(213,231)(214,227)(215,226)(216,228)
(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)(251,330)
(252,329)(253,335)(254,334)(255,336)(256,341)(257,340)(258,342)(259,338)
(260,337)(261,339)(262,345)(263,344)(264,343)(265,351)(266,350)(267,349)
(268,348)(269,347)(270,346)(271,401)(272,400)(273,402)(274,398)(275,397)
(276,399)(277,404)(278,403)(279,405)(280,384)(281,383)(282,382)(283,381)
(284,380)(285,379)(286,387)(287,386)(288,385)(289,391)(290,393)(291,392)
(292,388)(293,390)(294,389)(295,394)(296,396)(297,395)(298,366)(299,365)
(300,364)(301,363)(302,362)(303,361)(304,369)(305,368)(306,367)(307,373)
(308,375)(309,374)(310,370)(311,372)(312,371)(313,376)(314,378)(315,377)
(316,356)(317,355)(318,357)(319,353)(320,352)(321,354)(322,359)(323,358)
(324,360)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)(417,418)
(421,423)(424,427)(425,429)(426,428)(431,432)(433,486)(434,485)(435,484)
(436,483)(437,482)(438,481)(439,480)(440,479)(441,478)(442,466)(443,468)
(444,467)(445,463)(446,465)(447,464)(448,460)(449,462)(450,461)(451,476)
(452,475)(453,477)(454,473)(455,472)(456,474)(457,470)(458,469)(459,471);;
s2 := ( 1,271)( 2,273)( 3,272)( 4,277)( 5,279)( 6,278)( 7,274)( 8,276)
( 9,275)( 10,288)( 11,287)( 12,286)( 13,285)( 14,284)( 15,283)( 16,282)
( 17,281)( 18,280)( 19,293)( 20,292)( 21,294)( 22,290)( 23,289)( 24,291)
( 25,296)( 26,295)( 27,297)( 28,244)( 29,246)( 30,245)( 31,250)( 32,252)
( 33,251)( 34,247)( 35,249)( 36,248)( 37,261)( 38,260)( 39,259)( 40,258)
( 41,257)( 42,256)( 43,255)( 44,254)( 45,253)( 46,266)( 47,265)( 48,267)
( 49,263)( 50,262)( 51,264)( 52,269)( 53,268)( 54,270)( 55,300)( 56,299)
( 57,298)( 58,306)( 59,305)( 60,304)( 61,303)( 62,302)( 63,301)( 64,314)
( 65,313)( 66,315)( 67,311)( 68,310)( 69,312)( 70,308)( 71,307)( 72,309)
( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,322)( 80,324)
( 81,323)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)( 88,440)
( 89,439)( 90,441)( 91,442)( 92,444)( 93,443)( 94,448)( 95,450)( 96,449)
( 97,445)( 98,447)( 99,446)(100,459)(101,458)(102,457)(103,456)(104,455)
(105,454)(106,453)(107,452)(108,451)(109,410)(110,409)(111,411)(112,407)
(113,406)(114,408)(115,413)(116,412)(117,414)(118,415)(119,417)(120,416)
(121,421)(122,423)(123,422)(124,418)(125,420)(126,419)(127,432)(128,431)
(129,430)(130,429)(131,428)(132,427)(133,426)(134,425)(135,424)(136,463)
(137,465)(138,464)(139,460)(140,462)(141,461)(142,466)(143,468)(144,467)
(145,471)(146,470)(147,469)(148,477)(149,476)(150,475)(151,474)(152,473)
(153,472)(154,485)(155,484)(156,486)(157,482)(158,481)(159,483)(160,479)
(161,478)(162,480)(163,356)(164,355)(165,357)(166,353)(167,352)(168,354)
(169,359)(170,358)(171,360)(172,361)(173,363)(174,362)(175,367)(176,369)
(177,368)(178,364)(179,366)(180,365)(181,378)(182,377)(183,376)(184,375)
(185,374)(186,373)(187,372)(188,371)(189,370)(190,329)(191,328)(192,330)
(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,334)(200,336)
(201,335)(202,340)(203,342)(204,341)(205,337)(206,339)(207,338)(208,351)
(209,350)(210,349)(211,348)(212,347)(213,346)(214,345)(215,344)(216,343)
(217,382)(218,384)(219,383)(220,379)(221,381)(222,380)(223,385)(224,387)
(225,386)(226,390)(227,389)(228,388)(229,396)(230,395)(231,394)(232,393)
(233,392)(234,391)(235,404)(236,403)(237,405)(238,401)(239,400)(240,402)
(241,398)(242,397)(243,399);;
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, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2,
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(486)!( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)
( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)
( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)
( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)
( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)
( 85,165)( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)
( 93,184)( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)
(101,177)(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)
(109,194)(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)
(117,198)(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)
(125,215)(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)
(133,205)(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)
(141,218)(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)
(149,235)(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)
(157,228)(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)
(249,250)(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)
(260,265)(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)
(283,296)(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)
(303,304)(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)
(314,319)(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)
(331,412)(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)
(339,425)(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)
(347,415)(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)
(355,435)(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)
(363,454)(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)
(371,447)(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)
(379,464)(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)
(387,468)(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)
(395,485)(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)
(403,475)(404,476)(405,477);
s1 := Sym(486)!( 1, 82)( 2, 84)( 3, 83)( 4, 88)( 5, 90)( 6, 89)( 7, 85)
( 8, 87)( 9, 86)( 10, 92)( 11, 91)( 12, 93)( 13, 98)( 14, 97)( 15, 99)
( 16, 95)( 17, 94)( 18, 96)( 19,102)( 20,101)( 21,100)( 22,108)( 23,107)
( 24,106)( 25,105)( 26,104)( 27,103)( 28,158)( 29,157)( 30,159)( 31,155)
( 32,154)( 33,156)( 34,161)( 35,160)( 36,162)( 37,141)( 38,140)( 39,139)
( 40,138)( 41,137)( 42,136)( 43,144)( 44,143)( 45,142)( 46,148)( 47,150)
( 48,149)( 49,145)( 50,147)( 51,146)( 52,151)( 53,153)( 54,152)( 55,123)
( 56,122)( 57,121)( 58,120)( 59,119)( 60,118)( 61,126)( 62,125)( 63,124)
( 64,130)( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,133)( 71,135)
( 72,134)( 73,113)( 74,112)( 75,114)( 76,110)( 77,109)( 78,111)( 79,116)
( 80,115)( 81,117)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)
(174,175)(178,180)(181,184)(182,186)(183,185)(188,189)(190,243)(191,242)
(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,223)
(200,225)(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)
(208,233)(209,232)(210,234)(211,230)(212,229)(213,231)(214,227)(215,226)
(216,228)(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)
(251,330)(252,329)(253,335)(254,334)(255,336)(256,341)(257,340)(258,342)
(259,338)(260,337)(261,339)(262,345)(263,344)(264,343)(265,351)(266,350)
(267,349)(268,348)(269,347)(270,346)(271,401)(272,400)(273,402)(274,398)
(275,397)(276,399)(277,404)(278,403)(279,405)(280,384)(281,383)(282,382)
(283,381)(284,380)(285,379)(286,387)(287,386)(288,385)(289,391)(290,393)
(291,392)(292,388)(293,390)(294,389)(295,394)(296,396)(297,395)(298,366)
(299,365)(300,364)(301,363)(302,362)(303,361)(304,369)(305,368)(306,367)
(307,373)(308,375)(309,374)(310,370)(311,372)(312,371)(313,376)(314,378)
(315,377)(316,356)(317,355)(318,357)(319,353)(320,352)(321,354)(322,359)
(323,358)(324,360)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)
(417,418)(421,423)(424,427)(425,429)(426,428)(431,432)(433,486)(434,485)
(435,484)(436,483)(437,482)(438,481)(439,480)(440,479)(441,478)(442,466)
(443,468)(444,467)(445,463)(446,465)(447,464)(448,460)(449,462)(450,461)
(451,476)(452,475)(453,477)(454,473)(455,472)(456,474)(457,470)(458,469)
(459,471);
s2 := Sym(486)!( 1,271)( 2,273)( 3,272)( 4,277)( 5,279)( 6,278)( 7,274)
( 8,276)( 9,275)( 10,288)( 11,287)( 12,286)( 13,285)( 14,284)( 15,283)
( 16,282)( 17,281)( 18,280)( 19,293)( 20,292)( 21,294)( 22,290)( 23,289)
( 24,291)( 25,296)( 26,295)( 27,297)( 28,244)( 29,246)( 30,245)( 31,250)
( 32,252)( 33,251)( 34,247)( 35,249)( 36,248)( 37,261)( 38,260)( 39,259)
( 40,258)( 41,257)( 42,256)( 43,255)( 44,254)( 45,253)( 46,266)( 47,265)
( 48,267)( 49,263)( 50,262)( 51,264)( 52,269)( 53,268)( 54,270)( 55,300)
( 56,299)( 57,298)( 58,306)( 59,305)( 60,304)( 61,303)( 62,302)( 63,301)
( 64,314)( 65,313)( 66,315)( 67,311)( 68,310)( 69,312)( 70,308)( 71,307)
( 72,309)( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,322)
( 80,324)( 81,323)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)
( 88,440)( 89,439)( 90,441)( 91,442)( 92,444)( 93,443)( 94,448)( 95,450)
( 96,449)( 97,445)( 98,447)( 99,446)(100,459)(101,458)(102,457)(103,456)
(104,455)(105,454)(106,453)(107,452)(108,451)(109,410)(110,409)(111,411)
(112,407)(113,406)(114,408)(115,413)(116,412)(117,414)(118,415)(119,417)
(120,416)(121,421)(122,423)(123,422)(124,418)(125,420)(126,419)(127,432)
(128,431)(129,430)(130,429)(131,428)(132,427)(133,426)(134,425)(135,424)
(136,463)(137,465)(138,464)(139,460)(140,462)(141,461)(142,466)(143,468)
(144,467)(145,471)(146,470)(147,469)(148,477)(149,476)(150,475)(151,474)
(152,473)(153,472)(154,485)(155,484)(156,486)(157,482)(158,481)(159,483)
(160,479)(161,478)(162,480)(163,356)(164,355)(165,357)(166,353)(167,352)
(168,354)(169,359)(170,358)(171,360)(172,361)(173,363)(174,362)(175,367)
(176,369)(177,368)(178,364)(179,366)(180,365)(181,378)(182,377)(183,376)
(184,375)(185,374)(186,373)(187,372)(188,371)(189,370)(190,329)(191,328)
(192,330)(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,334)
(200,336)(201,335)(202,340)(203,342)(204,341)(205,337)(206,339)(207,338)
(208,351)(209,350)(210,349)(211,348)(212,347)(213,346)(214,345)(215,344)
(216,343)(217,382)(218,384)(219,383)(220,379)(221,381)(222,380)(223,385)
(224,387)(225,386)(226,390)(227,389)(228,388)(229,396)(230,395)(231,394)
(232,393)(233,392)(234,391)(235,404)(236,403)(237,405)(238,401)(239,400)
(240,402)(241,398)(242,397)(243,399);
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, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2,
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 >;
References : None.
to this polytope