Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944m
if this polytope has a name.
Group : SmallGroup(1944,946)
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
Self-Dual
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}*972e
3-fold quotients : {6,18}*648f, {18,6}*648f
6-fold quotients : {6,18}*324b, {18,6}*324b
9-fold quotients : {6,6}*216b
18-fold quotients : {6,6}*108
27-fold quotients : {6,6}*72a
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, 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, 93)( 11, 92)( 12, 91)( 13, 99)( 14, 98)( 15, 97)( 16, 96)
( 17, 95)( 18, 94)( 19,101)( 20,100)( 21,102)( 22,107)( 23,106)( 24,108)
( 25,104)( 26,103)( 27,105)( 28,162)( 29,161)( 30,160)( 31,159)( 32,158)
( 33,157)( 34,156)( 35,155)( 36,154)( 37,143)( 38,142)( 39,144)( 40,140)
( 41,139)( 42,141)( 43,137)( 44,136)( 45,138)( 46,151)( 47,153)( 48,152)
( 49,148)( 50,150)( 51,149)( 52,145)( 53,147)( 54,146)( 55,125)( 56,124)
( 57,126)( 58,122)( 59,121)( 60,123)( 61,119)( 62,118)( 63,120)( 64,133)
( 65,135)( 66,134)( 67,130)( 68,132)( 69,131)( 70,127)( 71,129)( 72,128)
( 73,117)( 74,116)( 75,115)( 76,114)( 77,113)( 78,112)( 79,111)( 80,110)
( 81,109)(163,167)(164,166)(165,168)(169,170)(172,175)(173,177)(174,176)
(179,180)(181,186)(182,185)(183,184)(187,189)(190,235)(191,237)(192,236)
(193,241)(194,243)(195,242)(196,238)(197,240)(198,239)(199,219)(200,218)
(201,217)(202,225)(203,224)(204,223)(205,222)(206,221)(207,220)(208,227)
(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)(216,231)
(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)(251,330)
(252,329)(253,336)(254,335)(255,334)(256,342)(257,341)(258,340)(259,339)
(260,338)(261,337)(262,344)(263,343)(264,345)(265,350)(266,349)(267,351)
(268,347)(269,346)(270,348)(271,405)(272,404)(273,403)(274,402)(275,401)
(276,400)(277,399)(278,398)(279,397)(280,386)(281,385)(282,387)(283,383)
(284,382)(285,384)(286,380)(287,379)(288,381)(289,394)(290,396)(291,395)
(292,391)(293,393)(294,392)(295,388)(296,390)(297,389)(298,368)(299,367)
(300,369)(301,365)(302,364)(303,366)(304,362)(305,361)(306,363)(307,376)
(308,378)(309,377)(310,373)(311,375)(312,374)(313,370)(314,372)(315,371)
(316,360)(317,359)(318,358)(319,357)(320,356)(321,355)(322,354)(323,353)
(324,352)(406,410)(407,409)(408,411)(412,413)(415,418)(416,420)(417,419)
(422,423)(424,429)(425,428)(426,427)(430,432)(433,478)(434,480)(435,479)
(436,484)(437,486)(438,485)(439,481)(440,483)(441,482)(442,462)(443,461)
(444,460)(445,468)(446,467)(447,466)(448,465)(449,464)(450,463)(451,470)
(452,469)(453,471)(454,476)(455,475)(456,477)(457,473)(458,472)(459,474);;
s2 := ( 1,271)( 2,273)( 3,272)( 4,276)( 5,275)( 6,274)( 7,278)( 8,277)
( 9,279)( 10,297)( 11,296)( 12,295)( 13,290)( 14,289)( 15,291)( 16,292)
( 17,294)( 18,293)( 19,284)( 20,283)( 21,285)( 22,286)( 23,288)( 24,287)
( 25,282)( 26,281)( 27,280)( 28,244)( 29,246)( 30,245)( 31,249)( 32,248)
( 33,247)( 34,251)( 35,250)( 36,252)( 37,270)( 38,269)( 39,268)( 40,263)
( 41,262)( 42,264)( 43,265)( 44,267)( 45,266)( 46,257)( 47,256)( 48,258)
( 49,259)( 50,261)( 51,260)( 52,255)( 53,254)( 54,253)( 55,299)( 56,298)
( 57,300)( 58,301)( 59,303)( 60,302)( 61,306)( 62,305)( 63,304)( 64,322)
( 65,324)( 66,323)( 67,318)( 68,317)( 69,316)( 70,320)( 71,319)( 72,321)
( 73,312)( 74,311)( 75,310)( 76,314)( 77,313)( 78,315)( 79,307)( 80,309)
( 81,308)( 82,352)( 83,354)( 84,353)( 85,357)( 86,356)( 87,355)( 88,359)
( 89,358)( 90,360)( 91,378)( 92,377)( 93,376)( 94,371)( 95,370)( 96,372)
( 97,373)( 98,375)( 99,374)(100,365)(101,364)(102,366)(103,367)(104,369)
(105,368)(106,363)(107,362)(108,361)(109,325)(110,327)(111,326)(112,330)
(113,329)(114,328)(115,332)(116,331)(117,333)(118,351)(119,350)(120,349)
(121,344)(122,343)(123,345)(124,346)(125,348)(126,347)(127,338)(128,337)
(129,339)(130,340)(131,342)(132,341)(133,336)(134,335)(135,334)(136,380)
(137,379)(138,381)(139,382)(140,384)(141,383)(142,387)(143,386)(144,385)
(145,403)(146,405)(147,404)(148,399)(149,398)(150,397)(151,401)(152,400)
(153,402)(154,393)(155,392)(156,391)(157,395)(158,394)(159,396)(160,388)
(161,390)(162,389)(163,433)(164,435)(165,434)(166,438)(167,437)(168,436)
(169,440)(170,439)(171,441)(172,459)(173,458)(174,457)(175,452)(176,451)
(177,453)(178,454)(179,456)(180,455)(181,446)(182,445)(183,447)(184,448)
(185,450)(186,449)(187,444)(188,443)(189,442)(190,406)(191,408)(192,407)
(193,411)(194,410)(195,409)(196,413)(197,412)(198,414)(199,432)(200,431)
(201,430)(202,425)(203,424)(204,426)(205,427)(206,429)(207,428)(208,419)
(209,418)(210,420)(211,421)(212,423)(213,422)(214,417)(215,416)(216,415)
(217,461)(218,460)(219,462)(220,463)(221,465)(222,464)(223,468)(224,467)
(225,466)(226,484)(227,486)(228,485)(229,480)(230,479)(231,478)(232,482)
(233,481)(234,483)(235,474)(236,473)(237,472)(238,476)(239,475)(240,477)
(241,469)(242,471)(243,470);;
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*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, 93)( 11, 92)( 12, 91)( 13, 99)( 14, 98)( 15, 97)
( 16, 96)( 17, 95)( 18, 94)( 19,101)( 20,100)( 21,102)( 22,107)( 23,106)
( 24,108)( 25,104)( 26,103)( 27,105)( 28,162)( 29,161)( 30,160)( 31,159)
( 32,158)( 33,157)( 34,156)( 35,155)( 36,154)( 37,143)( 38,142)( 39,144)
( 40,140)( 41,139)( 42,141)( 43,137)( 44,136)( 45,138)( 46,151)( 47,153)
( 48,152)( 49,148)( 50,150)( 51,149)( 52,145)( 53,147)( 54,146)( 55,125)
( 56,124)( 57,126)( 58,122)( 59,121)( 60,123)( 61,119)( 62,118)( 63,120)
( 64,133)( 65,135)( 66,134)( 67,130)( 68,132)( 69,131)( 70,127)( 71,129)
( 72,128)( 73,117)( 74,116)( 75,115)( 76,114)( 77,113)( 78,112)( 79,111)
( 80,110)( 81,109)(163,167)(164,166)(165,168)(169,170)(172,175)(173,177)
(174,176)(179,180)(181,186)(182,185)(183,184)(187,189)(190,235)(191,237)
(192,236)(193,241)(194,243)(195,242)(196,238)(197,240)(198,239)(199,219)
(200,218)(201,217)(202,225)(203,224)(204,223)(205,222)(206,221)(207,220)
(208,227)(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)
(216,231)(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)
(251,330)(252,329)(253,336)(254,335)(255,334)(256,342)(257,341)(258,340)
(259,339)(260,338)(261,337)(262,344)(263,343)(264,345)(265,350)(266,349)
(267,351)(268,347)(269,346)(270,348)(271,405)(272,404)(273,403)(274,402)
(275,401)(276,400)(277,399)(278,398)(279,397)(280,386)(281,385)(282,387)
(283,383)(284,382)(285,384)(286,380)(287,379)(288,381)(289,394)(290,396)
(291,395)(292,391)(293,393)(294,392)(295,388)(296,390)(297,389)(298,368)
(299,367)(300,369)(301,365)(302,364)(303,366)(304,362)(305,361)(306,363)
(307,376)(308,378)(309,377)(310,373)(311,375)(312,374)(313,370)(314,372)
(315,371)(316,360)(317,359)(318,358)(319,357)(320,356)(321,355)(322,354)
(323,353)(324,352)(406,410)(407,409)(408,411)(412,413)(415,418)(416,420)
(417,419)(422,423)(424,429)(425,428)(426,427)(430,432)(433,478)(434,480)
(435,479)(436,484)(437,486)(438,485)(439,481)(440,483)(441,482)(442,462)
(443,461)(444,460)(445,468)(446,467)(447,466)(448,465)(449,464)(450,463)
(451,470)(452,469)(453,471)(454,476)(455,475)(456,477)(457,473)(458,472)
(459,474);
s2 := Sym(486)!( 1,271)( 2,273)( 3,272)( 4,276)( 5,275)( 6,274)( 7,278)
( 8,277)( 9,279)( 10,297)( 11,296)( 12,295)( 13,290)( 14,289)( 15,291)
( 16,292)( 17,294)( 18,293)( 19,284)( 20,283)( 21,285)( 22,286)( 23,288)
( 24,287)( 25,282)( 26,281)( 27,280)( 28,244)( 29,246)( 30,245)( 31,249)
( 32,248)( 33,247)( 34,251)( 35,250)( 36,252)( 37,270)( 38,269)( 39,268)
( 40,263)( 41,262)( 42,264)( 43,265)( 44,267)( 45,266)( 46,257)( 47,256)
( 48,258)( 49,259)( 50,261)( 51,260)( 52,255)( 53,254)( 54,253)( 55,299)
( 56,298)( 57,300)( 58,301)( 59,303)( 60,302)( 61,306)( 62,305)( 63,304)
( 64,322)( 65,324)( 66,323)( 67,318)( 68,317)( 69,316)( 70,320)( 71,319)
( 72,321)( 73,312)( 74,311)( 75,310)( 76,314)( 77,313)( 78,315)( 79,307)
( 80,309)( 81,308)( 82,352)( 83,354)( 84,353)( 85,357)( 86,356)( 87,355)
( 88,359)( 89,358)( 90,360)( 91,378)( 92,377)( 93,376)( 94,371)( 95,370)
( 96,372)( 97,373)( 98,375)( 99,374)(100,365)(101,364)(102,366)(103,367)
(104,369)(105,368)(106,363)(107,362)(108,361)(109,325)(110,327)(111,326)
(112,330)(113,329)(114,328)(115,332)(116,331)(117,333)(118,351)(119,350)
(120,349)(121,344)(122,343)(123,345)(124,346)(125,348)(126,347)(127,338)
(128,337)(129,339)(130,340)(131,342)(132,341)(133,336)(134,335)(135,334)
(136,380)(137,379)(138,381)(139,382)(140,384)(141,383)(142,387)(143,386)
(144,385)(145,403)(146,405)(147,404)(148,399)(149,398)(150,397)(151,401)
(152,400)(153,402)(154,393)(155,392)(156,391)(157,395)(158,394)(159,396)
(160,388)(161,390)(162,389)(163,433)(164,435)(165,434)(166,438)(167,437)
(168,436)(169,440)(170,439)(171,441)(172,459)(173,458)(174,457)(175,452)
(176,451)(177,453)(178,454)(179,456)(180,455)(181,446)(182,445)(183,447)
(184,448)(185,450)(186,449)(187,444)(188,443)(189,442)(190,406)(191,408)
(192,407)(193,411)(194,410)(195,409)(196,413)(197,412)(198,414)(199,432)
(200,431)(201,430)(202,425)(203,424)(204,426)(205,427)(206,429)(207,428)
(208,419)(209,418)(210,420)(211,421)(212,423)(213,422)(214,417)(215,416)
(216,415)(217,461)(218,460)(219,462)(220,463)(221,465)(222,464)(223,468)
(224,467)(225,466)(226,484)(227,486)(228,485)(229,480)(230,479)(231,478)
(232,482)(233,481)(234,483)(235,474)(236,473)(237,472)(238,476)(239,475)
(240,477)(241,469)(242,471)(243,470);
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1 >;
References : None.
to this polytope