Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944c
if this polytope has a name.
Group : SmallGroup(1944,940)
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}*972a
3-fold quotients : {18,18}*648a, {6,18}*648b, {18,6}*648b
6-fold quotients : {6,18}*324a, {18,6}*324a
9-fold quotients : {6,18}*216a, {18,6}*216a, {6,6}*216b
18-fold quotients : {6,6}*108
27-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a
54-fold quotients : {2,9}*36, {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 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 82,172)( 83,174)( 84,173)( 85,175)
( 86,177)( 87,176)( 88,178)( 89,180)( 90,179)( 91,163)( 92,165)( 93,164)
( 94,166)( 95,168)( 96,167)( 97,169)( 98,171)( 99,170)(100,181)(101,183)
(102,182)(103,184)(104,186)(105,185)(106,187)(107,189)(108,188)(109,199)
(110,201)(111,200)(112,202)(113,204)(114,203)(115,205)(116,207)(117,206)
(118,190)(119,192)(120,191)(121,193)(122,195)(123,194)(124,196)(125,198)
(126,197)(127,208)(128,210)(129,209)(130,211)(131,213)(132,212)(133,214)
(134,216)(135,215)(136,226)(137,228)(138,227)(139,229)(140,231)(141,230)
(142,232)(143,234)(144,233)(145,217)(146,219)(147,218)(148,220)(149,222)
(150,221)(151,223)(152,225)(153,224)(154,235)(155,237)(156,236)(157,238)
(158,240)(159,239)(160,241)(161,243)(162,242)(245,246)(248,249)(251,252)
(253,262)(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)(260,270)
(261,269)(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)(283,292)
(284,294)(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)(305,306)
(307,316)(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)
(315,323)(325,415)(326,417)(327,416)(328,418)(329,420)(330,419)(331,421)
(332,423)(333,422)(334,406)(335,408)(336,407)(337,409)(338,411)(339,410)
(340,412)(341,414)(342,413)(343,424)(344,426)(345,425)(346,427)(347,429)
(348,428)(349,430)(350,432)(351,431)(352,442)(353,444)(354,443)(355,445)
(356,447)(357,446)(358,448)(359,450)(360,449)(361,433)(362,435)(363,434)
(364,436)(365,438)(366,437)(367,439)(368,441)(369,440)(370,451)(371,453)
(372,452)(373,454)(374,456)(375,455)(376,457)(377,459)(378,458)(379,469)
(380,471)(381,470)(382,472)(383,474)(384,473)(385,475)(386,477)(387,476)
(388,460)(389,462)(390,461)(391,463)(392,465)(393,464)(394,466)(395,468)
(396,467)(397,478)(398,480)(399,479)(400,481)(401,483)(402,482)(403,484)
(404,486)(405,485);;
s1 := ( 1, 82)( 2, 83)( 3, 84)( 4, 88)( 5, 89)( 6, 90)( 7, 85)( 8, 86)
( 9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)( 16,103)
( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)( 24, 99)
( 25, 94)( 26, 95)( 27, 96)( 28,141)( 29,139)( 30,140)( 31,138)( 32,136)
( 33,137)( 34,144)( 35,142)( 36,143)( 37,159)( 38,157)( 39,158)( 40,156)
( 41,154)( 42,155)( 43,162)( 44,160)( 45,161)( 46,150)( 47,148)( 48,149)
( 49,147)( 50,145)( 51,146)( 52,153)( 53,151)( 54,152)( 55,113)( 56,114)
( 57,112)( 58,110)( 59,111)( 60,109)( 61,116)( 62,117)( 63,115)( 64,131)
( 65,132)( 66,130)( 67,128)( 68,129)( 69,127)( 70,134)( 71,135)( 72,133)
( 73,122)( 74,123)( 75,121)( 76,119)( 77,120)( 78,118)( 79,125)( 80,126)
( 81,124)(163,172)(164,173)(165,174)(166,178)(167,179)(168,180)(169,175)
(170,176)(171,177)(184,187)(185,188)(186,189)(190,231)(191,229)(192,230)
(193,228)(194,226)(195,227)(196,234)(197,232)(198,233)(199,222)(200,220)
(201,221)(202,219)(203,217)(204,218)(205,225)(206,223)(207,224)(208,240)
(209,238)(210,239)(211,237)(212,235)(213,236)(214,243)(215,241)(216,242)
(244,325)(245,326)(246,327)(247,331)(248,332)(249,333)(250,328)(251,329)
(252,330)(253,343)(254,344)(255,345)(256,349)(257,350)(258,351)(259,346)
(260,347)(261,348)(262,334)(263,335)(264,336)(265,340)(266,341)(267,342)
(268,337)(269,338)(270,339)(271,384)(272,382)(273,383)(274,381)(275,379)
(276,380)(277,387)(278,385)(279,386)(280,402)(281,400)(282,401)(283,399)
(284,397)(285,398)(286,405)(287,403)(288,404)(289,393)(290,391)(291,392)
(292,390)(293,388)(294,389)(295,396)(296,394)(297,395)(298,356)(299,357)
(300,355)(301,353)(302,354)(303,352)(304,359)(305,360)(306,358)(307,374)
(308,375)(309,373)(310,371)(311,372)(312,370)(313,377)(314,378)(315,376)
(316,365)(317,366)(318,364)(319,362)(320,363)(321,361)(322,368)(323,369)
(324,367)(406,415)(407,416)(408,417)(409,421)(410,422)(411,423)(412,418)
(413,419)(414,420)(427,430)(428,431)(429,432)(433,474)(434,472)(435,473)
(436,471)(437,469)(438,470)(439,477)(440,475)(441,476)(442,465)(443,463)
(444,464)(445,462)(446,460)(447,461)(448,468)(449,466)(450,467)(451,483)
(452,481)(453,482)(454,480)(455,478)(456,479)(457,486)(458,484)(459,485);;
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)( 55,301)( 56,303)
( 57,302)( 58,298)( 59,300)( 60,299)( 61,304)( 62,306)( 63,305)( 64,310)
( 65,312)( 66,311)( 67,307)( 68,309)( 69,308)( 70,313)( 71,315)( 72,314)
( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,322)( 80,324)
( 81,323)( 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)(136,382)
(137,384)(138,383)(139,379)(140,381)(141,380)(142,385)(143,387)(144,386)
(145,391)(146,393)(147,392)(148,388)(149,390)(150,389)(151,394)(152,396)
(153,395)(154,400)(155,402)(156,401)(157,397)(158,399)(159,398)(160,403)
(161,405)(162,404)(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)
(217,463)(218,465)(219,464)(220,460)(221,462)(222,461)(223,466)(224,468)
(225,467)(226,472)(227,474)(228,473)(229,469)(230,471)(231,470)(232,475)
(233,477)(234,476)(235,481)(236,483)(237,482)(238,478)(239,480)(240,479)
(241,484)(242,486)(243,485);;
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*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(486)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 82,172)( 83,174)( 84,173)
( 85,175)( 86,177)( 87,176)( 88,178)( 89,180)( 90,179)( 91,163)( 92,165)
( 93,164)( 94,166)( 95,168)( 96,167)( 97,169)( 98,171)( 99,170)(100,181)
(101,183)(102,182)(103,184)(104,186)(105,185)(106,187)(107,189)(108,188)
(109,199)(110,201)(111,200)(112,202)(113,204)(114,203)(115,205)(116,207)
(117,206)(118,190)(119,192)(120,191)(121,193)(122,195)(123,194)(124,196)
(125,198)(126,197)(127,208)(128,210)(129,209)(130,211)(131,213)(132,212)
(133,214)(134,216)(135,215)(136,226)(137,228)(138,227)(139,229)(140,231)
(141,230)(142,232)(143,234)(144,233)(145,217)(146,219)(147,218)(148,220)
(149,222)(150,221)(151,223)(152,225)(153,224)(154,235)(155,237)(156,236)
(157,238)(158,240)(159,239)(160,241)(161,243)(162,242)(245,246)(248,249)
(251,252)(253,262)(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)
(260,270)(261,269)(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)
(283,292)(284,294)(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)
(305,306)(307,316)(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)
(314,324)(315,323)(325,415)(326,417)(327,416)(328,418)(329,420)(330,419)
(331,421)(332,423)(333,422)(334,406)(335,408)(336,407)(337,409)(338,411)
(339,410)(340,412)(341,414)(342,413)(343,424)(344,426)(345,425)(346,427)
(347,429)(348,428)(349,430)(350,432)(351,431)(352,442)(353,444)(354,443)
(355,445)(356,447)(357,446)(358,448)(359,450)(360,449)(361,433)(362,435)
(363,434)(364,436)(365,438)(366,437)(367,439)(368,441)(369,440)(370,451)
(371,453)(372,452)(373,454)(374,456)(375,455)(376,457)(377,459)(378,458)
(379,469)(380,471)(381,470)(382,472)(383,474)(384,473)(385,475)(386,477)
(387,476)(388,460)(389,462)(390,461)(391,463)(392,465)(393,464)(394,466)
(395,468)(396,467)(397,478)(398,480)(399,479)(400,481)(401,483)(402,482)
(403,484)(404,486)(405,485);
s1 := Sym(486)!( 1, 82)( 2, 83)( 3, 84)( 4, 88)( 5, 89)( 6, 90)( 7, 85)
( 8, 86)( 9, 87)( 10,100)( 11,101)( 12,102)( 13,106)( 14,107)( 15,108)
( 16,103)( 17,104)( 18,105)( 19, 91)( 20, 92)( 21, 93)( 22, 97)( 23, 98)
( 24, 99)( 25, 94)( 26, 95)( 27, 96)( 28,141)( 29,139)( 30,140)( 31,138)
( 32,136)( 33,137)( 34,144)( 35,142)( 36,143)( 37,159)( 38,157)( 39,158)
( 40,156)( 41,154)( 42,155)( 43,162)( 44,160)( 45,161)( 46,150)( 47,148)
( 48,149)( 49,147)( 50,145)( 51,146)( 52,153)( 53,151)( 54,152)( 55,113)
( 56,114)( 57,112)( 58,110)( 59,111)( 60,109)( 61,116)( 62,117)( 63,115)
( 64,131)( 65,132)( 66,130)( 67,128)( 68,129)( 69,127)( 70,134)( 71,135)
( 72,133)( 73,122)( 74,123)( 75,121)( 76,119)( 77,120)( 78,118)( 79,125)
( 80,126)( 81,124)(163,172)(164,173)(165,174)(166,178)(167,179)(168,180)
(169,175)(170,176)(171,177)(184,187)(185,188)(186,189)(190,231)(191,229)
(192,230)(193,228)(194,226)(195,227)(196,234)(197,232)(198,233)(199,222)
(200,220)(201,221)(202,219)(203,217)(204,218)(205,225)(206,223)(207,224)
(208,240)(209,238)(210,239)(211,237)(212,235)(213,236)(214,243)(215,241)
(216,242)(244,325)(245,326)(246,327)(247,331)(248,332)(249,333)(250,328)
(251,329)(252,330)(253,343)(254,344)(255,345)(256,349)(257,350)(258,351)
(259,346)(260,347)(261,348)(262,334)(263,335)(264,336)(265,340)(266,341)
(267,342)(268,337)(269,338)(270,339)(271,384)(272,382)(273,383)(274,381)
(275,379)(276,380)(277,387)(278,385)(279,386)(280,402)(281,400)(282,401)
(283,399)(284,397)(285,398)(286,405)(287,403)(288,404)(289,393)(290,391)
(291,392)(292,390)(293,388)(294,389)(295,396)(296,394)(297,395)(298,356)
(299,357)(300,355)(301,353)(302,354)(303,352)(304,359)(305,360)(306,358)
(307,374)(308,375)(309,373)(310,371)(311,372)(312,370)(313,377)(314,378)
(315,376)(316,365)(317,366)(318,364)(319,362)(320,363)(321,361)(322,368)
(323,369)(324,367)(406,415)(407,416)(408,417)(409,421)(410,422)(411,423)
(412,418)(413,419)(414,420)(427,430)(428,431)(429,432)(433,474)(434,472)
(435,473)(436,471)(437,469)(438,470)(439,477)(440,475)(441,476)(442,465)
(443,463)(444,464)(445,462)(446,460)(447,461)(448,468)(449,466)(450,467)
(451,483)(452,481)(453,482)(454,480)(455,478)(456,479)(457,486)(458,484)
(459,485);
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)( 55,301)
( 56,303)( 57,302)( 58,298)( 59,300)( 60,299)( 61,304)( 62,306)( 63,305)
( 64,310)( 65,312)( 66,311)( 67,307)( 68,309)( 69,308)( 70,313)( 71,315)
( 72,314)( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,322)
( 80,324)( 81,323)( 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)
(136,382)(137,384)(138,383)(139,379)(140,381)(141,380)(142,385)(143,387)
(144,386)(145,391)(146,393)(147,392)(148,388)(149,390)(150,389)(151,394)
(152,396)(153,395)(154,400)(155,402)(156,401)(157,397)(158,399)(159,398)
(160,403)(161,405)(162,404)(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)(217,463)(218,465)(219,464)(220,460)(221,462)(222,461)(223,466)
(224,468)(225,467)(226,472)(227,474)(228,473)(229,469)(230,471)(231,470)
(232,475)(233,477)(234,476)(235,481)(236,483)(237,482)(238,478)(239,480)
(240,479)(241,484)(242,486)(243,485);
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*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope