Polytope of Type {6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1944c
if this polytope has a name.
Group : SmallGroup(1944,943)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 54, 486, 162
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,18}*972a
3-fold quotients : {6,6}*648b, {6,18}*648f
6-fold quotients : {6,6}*324a, {6,18}*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, 9)( 5, 7)( 6, 8)( 10, 19)( 11, 20)( 12, 21)( 13, 27)( 14, 25)
( 15, 26)( 16, 23)( 17, 24)( 18, 22)( 31, 36)( 32, 34)( 33, 35)( 37, 46)
( 38, 47)( 39, 48)( 40, 54)( 41, 52)( 42, 53)( 43, 50)( 44, 51)( 45, 49)
( 58, 63)( 59, 61)( 60, 62)( 64, 73)( 65, 74)( 66, 75)( 67, 81)( 68, 79)
( 69, 80)( 70, 77)( 71, 78)( 72, 76)( 82,163)( 83,164)( 84,165)( 85,171)
( 86,169)( 87,170)( 88,167)( 89,168)( 90,166)( 91,181)( 92,182)( 93,183)
( 94,189)( 95,187)( 96,188)( 97,185)( 98,186)( 99,184)(100,172)(101,173)
(102,174)(103,180)(104,178)(105,179)(106,176)(107,177)(108,175)(109,190)
(110,191)(111,192)(112,198)(113,196)(114,197)(115,194)(116,195)(117,193)
(118,208)(119,209)(120,210)(121,216)(122,214)(123,215)(124,212)(125,213)
(126,211)(127,199)(128,200)(129,201)(130,207)(131,205)(132,206)(133,203)
(134,204)(135,202)(136,217)(137,218)(138,219)(139,225)(140,223)(141,224)
(142,221)(143,222)(144,220)(145,235)(146,236)(147,237)(148,243)(149,241)
(150,242)(151,239)(152,240)(153,238)(154,226)(155,227)(156,228)(157,234)
(158,232)(159,233)(160,230)(161,231)(162,229)(247,252)(248,250)(249,251)
(253,262)(254,263)(255,264)(256,270)(257,268)(258,269)(259,266)(260,267)
(261,265)(274,279)(275,277)(276,278)(280,289)(281,290)(282,291)(283,297)
(284,295)(285,296)(286,293)(287,294)(288,292)(301,306)(302,304)(303,305)
(307,316)(308,317)(309,318)(310,324)(311,322)(312,323)(313,320)(314,321)
(315,319)(325,406)(326,407)(327,408)(328,414)(329,412)(330,413)(331,410)
(332,411)(333,409)(334,424)(335,425)(336,426)(337,432)(338,430)(339,431)
(340,428)(341,429)(342,427)(343,415)(344,416)(345,417)(346,423)(347,421)
(348,422)(349,419)(350,420)(351,418)(352,433)(353,434)(354,435)(355,441)
(356,439)(357,440)(358,437)(359,438)(360,436)(361,451)(362,452)(363,453)
(364,459)(365,457)(366,458)(367,455)(368,456)(369,454)(370,442)(371,443)
(372,444)(373,450)(374,448)(375,449)(376,446)(377,447)(378,445)(379,460)
(380,461)(381,462)(382,468)(383,466)(384,467)(385,464)(386,465)(387,463)
(388,478)(389,479)(390,480)(391,486)(392,484)(393,485)(394,482)(395,483)
(396,481)(397,469)(398,470)(399,471)(400,477)(401,475)(402,476)(403,473)
(404,474)(405,472);;
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)( 28,159)( 29,158)( 30,157)( 31,156)( 32,155)
( 33,154)( 34,162)( 35,161)( 36,160)( 37,140)( 38,139)( 39,141)( 40,137)
( 41,136)( 42,138)( 43,143)( 44,142)( 45,144)( 46,148)( 47,150)( 48,149)
( 49,145)( 50,147)( 51,146)( 52,151)( 53,153)( 54,152)( 55,122)( 56,121)
( 57,123)( 58,119)( 59,118)( 60,120)( 61,125)( 62,124)( 63,126)( 64,130)
( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,133)( 71,135)( 72,134)
( 73,114)( 74,113)( 75,112)( 76,111)( 77,110)( 78,109)( 79,117)( 80,116)
( 81,115)(164,165)(166,169)(167,171)(168,170)(172,174)(175,180)(176,179)
(177,178)(181,182)(184,188)(185,187)(186,189)(190,240)(191,239)(192,238)
(193,237)(194,236)(195,235)(196,243)(197,242)(198,241)(199,221)(200,220)
(201,222)(202,218)(203,217)(204,219)(205,224)(206,223)(207,225)(208,229)
(209,231)(210,230)(211,226)(212,228)(213,227)(214,232)(215,234)(216,233)
(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)(271,402)(272,401)(273,400)(274,399)(275,398)
(276,397)(277,405)(278,404)(279,403)(280,383)(281,382)(282,384)(283,380)
(284,379)(285,381)(286,386)(287,385)(288,387)(289,391)(290,393)(291,392)
(292,388)(293,390)(294,389)(295,394)(296,396)(297,395)(298,365)(299,364)
(300,366)(301,362)(302,361)(303,363)(304,368)(305,367)(306,369)(307,373)
(308,375)(309,374)(310,370)(311,372)(312,371)(313,376)(314,378)(315,377)
(316,357)(317,356)(318,355)(319,354)(320,353)(321,352)(322,360)(323,359)
(324,358)(407,408)(409,412)(410,414)(411,413)(415,417)(418,423)(419,422)
(420,421)(424,425)(427,431)(428,430)(429,432)(433,483)(434,482)(435,481)
(436,480)(437,479)(438,478)(439,486)(440,485)(441,484)(442,464)(443,463)
(444,465)(445,461)(446,460)(447,462)(448,467)(449,466)(450,468)(451,472)
(452,474)(453,473)(454,469)(455,471)(456,470)(457,475)(458,477)(459,476);;
s2 := ( 1,271)( 2,273)( 3,272)( 4,275)( 5,274)( 6,276)( 7,279)( 8,278)
( 9,277)( 10,294)( 11,293)( 12,292)( 13,295)( 14,297)( 15,296)( 16,290)
( 17,289)( 18,291)( 19,287)( 20,286)( 21,288)( 22,282)( 23,281)( 24,280)
( 25,283)( 26,285)( 27,284)( 28,244)( 29,246)( 30,245)( 31,248)( 32,247)
( 33,249)( 34,252)( 35,251)( 36,250)( 37,267)( 38,266)( 39,265)( 40,268)
( 41,270)( 42,269)( 43,263)( 44,262)( 45,264)( 46,260)( 47,259)( 48,261)
( 49,255)( 50,254)( 51,253)( 52,256)( 53,258)( 54,257)( 55,299)( 56,298)
( 57,300)( 58,303)( 59,302)( 60,301)( 61,304)( 62,306)( 63,305)( 64,319)
( 65,321)( 66,320)( 67,323)( 68,322)( 69,324)( 70,318)( 71,317)( 72,316)
( 73,315)( 74,314)( 75,313)( 76,307)( 77,309)( 78,308)( 79,311)( 80,310)
( 81,312)( 82,352)( 83,354)( 84,353)( 85,356)( 86,355)( 87,357)( 88,360)
( 89,359)( 90,358)( 91,375)( 92,374)( 93,373)( 94,376)( 95,378)( 96,377)
( 97,371)( 98,370)( 99,372)(100,368)(101,367)(102,369)(103,363)(104,362)
(105,361)(106,364)(107,366)(108,365)(109,325)(110,327)(111,326)(112,329)
(113,328)(114,330)(115,333)(116,332)(117,331)(118,348)(119,347)(120,346)
(121,349)(122,351)(123,350)(124,344)(125,343)(126,345)(127,341)(128,340)
(129,342)(130,336)(131,335)(132,334)(133,337)(134,339)(135,338)(136,380)
(137,379)(138,381)(139,384)(140,383)(141,382)(142,385)(143,387)(144,386)
(145,400)(146,402)(147,401)(148,404)(149,403)(150,405)(151,399)(152,398)
(153,397)(154,396)(155,395)(156,394)(157,388)(158,390)(159,389)(160,392)
(161,391)(162,393)(163,433)(164,435)(165,434)(166,437)(167,436)(168,438)
(169,441)(170,440)(171,439)(172,456)(173,455)(174,454)(175,457)(176,459)
(177,458)(178,452)(179,451)(180,453)(181,449)(182,448)(183,450)(184,444)
(185,443)(186,442)(187,445)(188,447)(189,446)(190,406)(191,408)(192,407)
(193,410)(194,409)(195,411)(196,414)(197,413)(198,412)(199,429)(200,428)
(201,427)(202,430)(203,432)(204,431)(205,425)(206,424)(207,426)(208,422)
(209,421)(210,423)(211,417)(212,416)(213,415)(214,418)(215,420)(216,419)
(217,461)(218,460)(219,462)(220,465)(221,464)(222,463)(223,466)(224,468)
(225,467)(226,481)(227,483)(228,482)(229,485)(230,484)(231,486)(232,480)
(233,479)(234,478)(235,477)(236,476)(237,475)(238,469)(239,471)(240,470)
(241,473)(242,472)(243,474);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(486)!( 4, 9)( 5, 7)( 6, 8)( 10, 19)( 11, 20)( 12, 21)( 13, 27)
( 14, 25)( 15, 26)( 16, 23)( 17, 24)( 18, 22)( 31, 36)( 32, 34)( 33, 35)
( 37, 46)( 38, 47)( 39, 48)( 40, 54)( 41, 52)( 42, 53)( 43, 50)( 44, 51)
( 45, 49)( 58, 63)( 59, 61)( 60, 62)( 64, 73)( 65, 74)( 66, 75)( 67, 81)
( 68, 79)( 69, 80)( 70, 77)( 71, 78)( 72, 76)( 82,163)( 83,164)( 84,165)
( 85,171)( 86,169)( 87,170)( 88,167)( 89,168)( 90,166)( 91,181)( 92,182)
( 93,183)( 94,189)( 95,187)( 96,188)( 97,185)( 98,186)( 99,184)(100,172)
(101,173)(102,174)(103,180)(104,178)(105,179)(106,176)(107,177)(108,175)
(109,190)(110,191)(111,192)(112,198)(113,196)(114,197)(115,194)(116,195)
(117,193)(118,208)(119,209)(120,210)(121,216)(122,214)(123,215)(124,212)
(125,213)(126,211)(127,199)(128,200)(129,201)(130,207)(131,205)(132,206)
(133,203)(134,204)(135,202)(136,217)(137,218)(138,219)(139,225)(140,223)
(141,224)(142,221)(143,222)(144,220)(145,235)(146,236)(147,237)(148,243)
(149,241)(150,242)(151,239)(152,240)(153,238)(154,226)(155,227)(156,228)
(157,234)(158,232)(159,233)(160,230)(161,231)(162,229)(247,252)(248,250)
(249,251)(253,262)(254,263)(255,264)(256,270)(257,268)(258,269)(259,266)
(260,267)(261,265)(274,279)(275,277)(276,278)(280,289)(281,290)(282,291)
(283,297)(284,295)(285,296)(286,293)(287,294)(288,292)(301,306)(302,304)
(303,305)(307,316)(308,317)(309,318)(310,324)(311,322)(312,323)(313,320)
(314,321)(315,319)(325,406)(326,407)(327,408)(328,414)(329,412)(330,413)
(331,410)(332,411)(333,409)(334,424)(335,425)(336,426)(337,432)(338,430)
(339,431)(340,428)(341,429)(342,427)(343,415)(344,416)(345,417)(346,423)
(347,421)(348,422)(349,419)(350,420)(351,418)(352,433)(353,434)(354,435)
(355,441)(356,439)(357,440)(358,437)(359,438)(360,436)(361,451)(362,452)
(363,453)(364,459)(365,457)(366,458)(367,455)(368,456)(369,454)(370,442)
(371,443)(372,444)(373,450)(374,448)(375,449)(376,446)(377,447)(378,445)
(379,460)(380,461)(381,462)(382,468)(383,466)(384,467)(385,464)(386,465)
(387,463)(388,478)(389,479)(390,480)(391,486)(392,484)(393,485)(394,482)
(395,483)(396,481)(397,469)(398,470)(399,471)(400,477)(401,475)(402,476)
(403,473)(404,474)(405,472);
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)( 28,159)( 29,158)( 30,157)( 31,156)
( 32,155)( 33,154)( 34,162)( 35,161)( 36,160)( 37,140)( 38,139)( 39,141)
( 40,137)( 41,136)( 42,138)( 43,143)( 44,142)( 45,144)( 46,148)( 47,150)
( 48,149)( 49,145)( 50,147)( 51,146)( 52,151)( 53,153)( 54,152)( 55,122)
( 56,121)( 57,123)( 58,119)( 59,118)( 60,120)( 61,125)( 62,124)( 63,126)
( 64,130)( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,133)( 71,135)
( 72,134)( 73,114)( 74,113)( 75,112)( 76,111)( 77,110)( 78,109)( 79,117)
( 80,116)( 81,115)(164,165)(166,169)(167,171)(168,170)(172,174)(175,180)
(176,179)(177,178)(181,182)(184,188)(185,187)(186,189)(190,240)(191,239)
(192,238)(193,237)(194,236)(195,235)(196,243)(197,242)(198,241)(199,221)
(200,220)(201,222)(202,218)(203,217)(204,219)(205,224)(206,223)(207,225)
(208,229)(209,231)(210,230)(211,226)(212,228)(213,227)(214,232)(215,234)
(216,233)(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)(271,402)(272,401)(273,400)(274,399)
(275,398)(276,397)(277,405)(278,404)(279,403)(280,383)(281,382)(282,384)
(283,380)(284,379)(285,381)(286,386)(287,385)(288,387)(289,391)(290,393)
(291,392)(292,388)(293,390)(294,389)(295,394)(296,396)(297,395)(298,365)
(299,364)(300,366)(301,362)(302,361)(303,363)(304,368)(305,367)(306,369)
(307,373)(308,375)(309,374)(310,370)(311,372)(312,371)(313,376)(314,378)
(315,377)(316,357)(317,356)(318,355)(319,354)(320,353)(321,352)(322,360)
(323,359)(324,358)(407,408)(409,412)(410,414)(411,413)(415,417)(418,423)
(419,422)(420,421)(424,425)(427,431)(428,430)(429,432)(433,483)(434,482)
(435,481)(436,480)(437,479)(438,478)(439,486)(440,485)(441,484)(442,464)
(443,463)(444,465)(445,461)(446,460)(447,462)(448,467)(449,466)(450,468)
(451,472)(452,474)(453,473)(454,469)(455,471)(456,470)(457,475)(458,477)
(459,476);
s2 := Sym(486)!( 1,271)( 2,273)( 3,272)( 4,275)( 5,274)( 6,276)( 7,279)
( 8,278)( 9,277)( 10,294)( 11,293)( 12,292)( 13,295)( 14,297)( 15,296)
( 16,290)( 17,289)( 18,291)( 19,287)( 20,286)( 21,288)( 22,282)( 23,281)
( 24,280)( 25,283)( 26,285)( 27,284)( 28,244)( 29,246)( 30,245)( 31,248)
( 32,247)( 33,249)( 34,252)( 35,251)( 36,250)( 37,267)( 38,266)( 39,265)
( 40,268)( 41,270)( 42,269)( 43,263)( 44,262)( 45,264)( 46,260)( 47,259)
( 48,261)( 49,255)( 50,254)( 51,253)( 52,256)( 53,258)( 54,257)( 55,299)
( 56,298)( 57,300)( 58,303)( 59,302)( 60,301)( 61,304)( 62,306)( 63,305)
( 64,319)( 65,321)( 66,320)( 67,323)( 68,322)( 69,324)( 70,318)( 71,317)
( 72,316)( 73,315)( 74,314)( 75,313)( 76,307)( 77,309)( 78,308)( 79,311)
( 80,310)( 81,312)( 82,352)( 83,354)( 84,353)( 85,356)( 86,355)( 87,357)
( 88,360)( 89,359)( 90,358)( 91,375)( 92,374)( 93,373)( 94,376)( 95,378)
( 96,377)( 97,371)( 98,370)( 99,372)(100,368)(101,367)(102,369)(103,363)
(104,362)(105,361)(106,364)(107,366)(108,365)(109,325)(110,327)(111,326)
(112,329)(113,328)(114,330)(115,333)(116,332)(117,331)(118,348)(119,347)
(120,346)(121,349)(122,351)(123,350)(124,344)(125,343)(126,345)(127,341)
(128,340)(129,342)(130,336)(131,335)(132,334)(133,337)(134,339)(135,338)
(136,380)(137,379)(138,381)(139,384)(140,383)(141,382)(142,385)(143,387)
(144,386)(145,400)(146,402)(147,401)(148,404)(149,403)(150,405)(151,399)
(152,398)(153,397)(154,396)(155,395)(156,394)(157,388)(158,390)(159,389)
(160,392)(161,391)(162,393)(163,433)(164,435)(165,434)(166,437)(167,436)
(168,438)(169,441)(170,440)(171,439)(172,456)(173,455)(174,454)(175,457)
(176,459)(177,458)(178,452)(179,451)(180,453)(181,449)(182,448)(183,450)
(184,444)(185,443)(186,442)(187,445)(188,447)(189,446)(190,406)(191,408)
(192,407)(193,410)(194,409)(195,411)(196,414)(197,413)(198,412)(199,429)
(200,428)(201,427)(202,430)(203,432)(204,431)(205,425)(206,424)(207,426)
(208,422)(209,421)(210,423)(211,417)(212,416)(213,415)(214,418)(215,420)
(216,419)(217,461)(218,460)(219,462)(220,465)(221,464)(222,463)(223,466)
(224,468)(225,467)(226,481)(227,483)(228,482)(229,485)(230,484)(231,486)
(232,480)(233,479)(234,478)(235,477)(236,476)(237,475)(238,469)(239,471)
(240,470)(241,473)(242,472)(243,474);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
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*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1 >;
References : None.
to this polytope