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