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