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