Questions?
See the FAQ
or other info.

Polytope of Type {6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1944j
if this polytope has a name.
Group : SmallGroup(1944,952)
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}*972d
   3-fold quotients : {6,18}*648b, {6,6}*648a, {6,18}*648f, {6,18}*648g
   6-fold quotients : {6,18}*324a, {6,6}*324b, {6,18}*324b, {6,18}*324c
   9-fold quotients : {6,18}*216a, {6,6}*216b
   18-fold quotients : {6,6}*108
   27-fold quotients : {2,18}*72, {6,6}*72a
   54-fold quotients : {2,9}*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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)( 34, 58)
( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)( 42, 72)
( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)( 50, 80)
( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)(196,220)
(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)(204,234)
(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)(212,242)
(213,243)(214,238)(215,239)(216,240)(247,250)(248,251)(249,252)(256,259)
(257,260)(258,261)(265,268)(266,269)(267,270)(271,298)(272,299)(273,300)
(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)(281,308)
(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)(289,316)
(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)(297,321)
(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)(347,350)
(348,351)(352,379)(353,380)(354,381)(355,385)(356,386)(357,387)(358,382)
(359,383)(360,384)(361,388)(362,389)(363,390)(364,394)(365,395)(366,396)
(367,391)(368,392)(369,393)(370,397)(371,398)(372,399)(373,403)(374,404)
(375,405)(376,400)(377,401)(378,402)(409,412)(410,413)(411,414)(418,421)
(419,422)(420,423)(427,430)(428,431)(429,432)(433,460)(434,461)(435,462)
(436,466)(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)(443,470)
(444,471)(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)(451,478)
(452,479)(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)(459,483);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)
(  9, 36)( 10, 46)( 11, 48)( 12, 47)( 13, 51)( 14, 50)( 15, 49)( 16, 53)
( 17, 52)( 18, 54)( 19, 37)( 20, 39)( 21, 38)( 22, 42)( 23, 41)( 24, 40)
( 25, 44)( 26, 43)( 27, 45)( 56, 57)( 58, 60)( 61, 62)( 64, 73)( 65, 75)
( 66, 74)( 67, 78)( 68, 77)( 69, 76)( 70, 80)( 71, 79)( 72, 81)( 82,199)
( 83,201)( 84,200)( 85,204)( 86,203)( 87,202)( 88,206)( 89,205)( 90,207)
( 91,190)( 92,192)( 93,191)( 94,195)( 95,194)( 96,193)( 97,197)( 98,196)
( 99,198)(100,208)(101,210)(102,209)(103,213)(104,212)(105,211)(106,215)
(107,214)(108,216)(109,172)(110,174)(111,173)(112,177)(113,176)(114,175)
(115,179)(116,178)(117,180)(118,163)(119,165)(120,164)(121,168)(122,167)
(123,166)(124,170)(125,169)(126,171)(127,181)(128,183)(129,182)(130,186)
(131,185)(132,184)(133,188)(134,187)(135,189)(136,226)(137,228)(138,227)
(139,231)(140,230)(141,229)(142,233)(143,232)(144,234)(145,217)(146,219)
(147,218)(148,222)(149,221)(150,220)(151,224)(152,223)(153,225)(154,235)
(155,237)(156,236)(157,240)(158,239)(159,238)(160,242)(161,241)(162,243)
(244,271)(245,273)(246,272)(247,276)(248,275)(249,274)(250,278)(251,277)
(252,279)(253,289)(254,291)(255,290)(256,294)(257,293)(258,292)(259,296)
(260,295)(261,297)(262,280)(263,282)(264,281)(265,285)(266,284)(267,283)
(268,287)(269,286)(270,288)(299,300)(301,303)(304,305)(307,316)(308,318)
(309,317)(310,321)(311,320)(312,319)(313,323)(314,322)(315,324)(325,442)
(326,444)(327,443)(328,447)(329,446)(330,445)(331,449)(332,448)(333,450)
(334,433)(335,435)(336,434)(337,438)(338,437)(339,436)(340,440)(341,439)
(342,441)(343,451)(344,453)(345,452)(346,456)(347,455)(348,454)(349,458)
(350,457)(351,459)(352,415)(353,417)(354,416)(355,420)(356,419)(357,418)
(358,422)(359,421)(360,423)(361,406)(362,408)(363,407)(364,411)(365,410)
(366,409)(367,413)(368,412)(369,414)(370,424)(371,426)(372,425)(373,429)
(374,428)(375,427)(376,431)(377,430)(378,432)(379,469)(380,471)(381,470)
(382,474)(383,473)(384,472)(385,476)(386,475)(387,477)(388,460)(389,462)
(390,461)(391,465)(392,464)(393,463)(394,467)(395,466)(396,468)(397,478)
(398,480)(399,479)(400,483)(401,482)(402,481)(403,485)(404,484)(405,486);;
s2 := (  1,325)(  2,327)(  3,326)(  4,331)(  5,333)(  6,332)(  7,328)(  8,330)
(  9,329)( 10,343)( 11,345)( 12,344)( 13,349)( 14,351)( 15,350)( 16,346)
( 17,348)( 18,347)( 19,334)( 20,336)( 21,335)( 22,340)( 23,342)( 24,341)
( 25,337)( 26,339)( 27,338)( 28,356)( 29,355)( 30,357)( 31,353)( 32,352)
( 33,354)( 34,359)( 35,358)( 36,360)( 37,374)( 38,373)( 39,375)( 40,371)
( 41,370)( 42,372)( 43,377)( 44,376)( 45,378)( 46,365)( 47,364)( 48,366)
( 49,362)( 50,361)( 51,363)( 52,368)( 53,367)( 54,369)( 55,386)( 56,385)
( 57,387)( 58,383)( 59,382)( 60,384)( 61,380)( 62,379)( 63,381)( 64,404)
( 65,403)( 66,405)( 67,401)( 68,400)( 69,402)( 70,398)( 71,397)( 72,399)
( 73,395)( 74,394)( 75,396)( 76,392)( 77,391)( 78,393)( 79,389)( 80,388)
( 81,390)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)( 88,247)
( 89,249)( 90,248)( 91,262)( 92,264)( 93,263)( 94,268)( 95,270)( 96,269)
( 97,265)( 98,267)( 99,266)(100,253)(101,255)(102,254)(103,259)(104,261)
(105,260)(106,256)(107,258)(108,257)(109,275)(110,274)(111,276)(112,272)
(113,271)(114,273)(115,278)(116,277)(117,279)(118,293)(119,292)(120,294)
(121,290)(122,289)(123,291)(124,296)(125,295)(126,297)(127,284)(128,283)
(129,285)(130,281)(131,280)(132,282)(133,287)(134,286)(135,288)(136,305)
(137,304)(138,306)(139,302)(140,301)(141,303)(142,299)(143,298)(144,300)
(145,323)(146,322)(147,324)(148,320)(149,319)(150,321)(151,317)(152,316)
(153,318)(154,314)(155,313)(156,315)(157,311)(158,310)(159,312)(160,308)
(161,307)(162,309)(163,415)(164,417)(165,416)(166,421)(167,423)(168,422)
(169,418)(170,420)(171,419)(172,406)(173,408)(174,407)(175,412)(176,414)
(177,413)(178,409)(179,411)(180,410)(181,424)(182,426)(183,425)(184,430)
(185,432)(186,431)(187,427)(188,429)(189,428)(190,446)(191,445)(192,447)
(193,443)(194,442)(195,444)(196,449)(197,448)(198,450)(199,437)(200,436)
(201,438)(202,434)(203,433)(204,435)(205,440)(206,439)(207,441)(208,455)
(209,454)(210,456)(211,452)(212,451)(213,453)(214,458)(215,457)(216,459)
(217,476)(218,475)(219,477)(220,473)(221,472)(222,474)(223,470)(224,469)
(225,471)(226,467)(227,466)(228,468)(229,464)(230,463)(231,465)(232,461)
(233,460)(234,462)(235,485)(236,484)(237,486)(238,482)(239,481)(240,483)
(241,479)(242,478)(243,480);;
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*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)
( 34, 58)( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)
( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)
( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)
(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)
(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)
(135,159)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)
(196,220)(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)
(204,234)(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)
(212,242)(213,243)(214,238)(215,239)(216,240)(247,250)(248,251)(249,252)
(256,259)(257,260)(258,261)(265,268)(266,269)(267,270)(271,298)(272,299)
(273,300)(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)
(281,308)(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)
(289,316)(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)
(297,321)(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)
(347,350)(348,351)(352,379)(353,380)(354,381)(355,385)(356,386)(357,387)
(358,382)(359,383)(360,384)(361,388)(362,389)(363,390)(364,394)(365,395)
(366,396)(367,391)(368,392)(369,393)(370,397)(371,398)(372,399)(373,403)
(374,404)(375,405)(376,400)(377,401)(378,402)(409,412)(410,413)(411,414)
(418,421)(419,422)(420,423)(427,430)(428,431)(429,432)(433,460)(434,461)
(435,462)(436,466)(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)
(443,470)(444,471)(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)
(451,478)(452,479)(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)
(459,483);
s1 := Sym(486)!(  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)
(  8, 34)(  9, 36)( 10, 46)( 11, 48)( 12, 47)( 13, 51)( 14, 50)( 15, 49)
( 16, 53)( 17, 52)( 18, 54)( 19, 37)( 20, 39)( 21, 38)( 22, 42)( 23, 41)
( 24, 40)( 25, 44)( 26, 43)( 27, 45)( 56, 57)( 58, 60)( 61, 62)( 64, 73)
( 65, 75)( 66, 74)( 67, 78)( 68, 77)( 69, 76)( 70, 80)( 71, 79)( 72, 81)
( 82,199)( 83,201)( 84,200)( 85,204)( 86,203)( 87,202)( 88,206)( 89,205)
( 90,207)( 91,190)( 92,192)( 93,191)( 94,195)( 95,194)( 96,193)( 97,197)
( 98,196)( 99,198)(100,208)(101,210)(102,209)(103,213)(104,212)(105,211)
(106,215)(107,214)(108,216)(109,172)(110,174)(111,173)(112,177)(113,176)
(114,175)(115,179)(116,178)(117,180)(118,163)(119,165)(120,164)(121,168)
(122,167)(123,166)(124,170)(125,169)(126,171)(127,181)(128,183)(129,182)
(130,186)(131,185)(132,184)(133,188)(134,187)(135,189)(136,226)(137,228)
(138,227)(139,231)(140,230)(141,229)(142,233)(143,232)(144,234)(145,217)
(146,219)(147,218)(148,222)(149,221)(150,220)(151,224)(152,223)(153,225)
(154,235)(155,237)(156,236)(157,240)(158,239)(159,238)(160,242)(161,241)
(162,243)(244,271)(245,273)(246,272)(247,276)(248,275)(249,274)(250,278)
(251,277)(252,279)(253,289)(254,291)(255,290)(256,294)(257,293)(258,292)
(259,296)(260,295)(261,297)(262,280)(263,282)(264,281)(265,285)(266,284)
(267,283)(268,287)(269,286)(270,288)(299,300)(301,303)(304,305)(307,316)
(308,318)(309,317)(310,321)(311,320)(312,319)(313,323)(314,322)(315,324)
(325,442)(326,444)(327,443)(328,447)(329,446)(330,445)(331,449)(332,448)
(333,450)(334,433)(335,435)(336,434)(337,438)(338,437)(339,436)(340,440)
(341,439)(342,441)(343,451)(344,453)(345,452)(346,456)(347,455)(348,454)
(349,458)(350,457)(351,459)(352,415)(353,417)(354,416)(355,420)(356,419)
(357,418)(358,422)(359,421)(360,423)(361,406)(362,408)(363,407)(364,411)
(365,410)(366,409)(367,413)(368,412)(369,414)(370,424)(371,426)(372,425)
(373,429)(374,428)(375,427)(376,431)(377,430)(378,432)(379,469)(380,471)
(381,470)(382,474)(383,473)(384,472)(385,476)(386,475)(387,477)(388,460)
(389,462)(390,461)(391,465)(392,464)(393,463)(394,467)(395,466)(396,468)
(397,478)(398,480)(399,479)(400,483)(401,482)(402,481)(403,485)(404,484)
(405,486);
s2 := Sym(486)!(  1,325)(  2,327)(  3,326)(  4,331)(  5,333)(  6,332)(  7,328)
(  8,330)(  9,329)( 10,343)( 11,345)( 12,344)( 13,349)( 14,351)( 15,350)
( 16,346)( 17,348)( 18,347)( 19,334)( 20,336)( 21,335)( 22,340)( 23,342)
( 24,341)( 25,337)( 26,339)( 27,338)( 28,356)( 29,355)( 30,357)( 31,353)
( 32,352)( 33,354)( 34,359)( 35,358)( 36,360)( 37,374)( 38,373)( 39,375)
( 40,371)( 41,370)( 42,372)( 43,377)( 44,376)( 45,378)( 46,365)( 47,364)
( 48,366)( 49,362)( 50,361)( 51,363)( 52,368)( 53,367)( 54,369)( 55,386)
( 56,385)( 57,387)( 58,383)( 59,382)( 60,384)( 61,380)( 62,379)( 63,381)
( 64,404)( 65,403)( 66,405)( 67,401)( 68,400)( 69,402)( 70,398)( 71,397)
( 72,399)( 73,395)( 74,394)( 75,396)( 76,392)( 77,391)( 78,393)( 79,389)
( 80,388)( 81,390)( 82,244)( 83,246)( 84,245)( 85,250)( 86,252)( 87,251)
( 88,247)( 89,249)( 90,248)( 91,262)( 92,264)( 93,263)( 94,268)( 95,270)
( 96,269)( 97,265)( 98,267)( 99,266)(100,253)(101,255)(102,254)(103,259)
(104,261)(105,260)(106,256)(107,258)(108,257)(109,275)(110,274)(111,276)
(112,272)(113,271)(114,273)(115,278)(116,277)(117,279)(118,293)(119,292)
(120,294)(121,290)(122,289)(123,291)(124,296)(125,295)(126,297)(127,284)
(128,283)(129,285)(130,281)(131,280)(132,282)(133,287)(134,286)(135,288)
(136,305)(137,304)(138,306)(139,302)(140,301)(141,303)(142,299)(143,298)
(144,300)(145,323)(146,322)(147,324)(148,320)(149,319)(150,321)(151,317)
(152,316)(153,318)(154,314)(155,313)(156,315)(157,311)(158,310)(159,312)
(160,308)(161,307)(162,309)(163,415)(164,417)(165,416)(166,421)(167,423)
(168,422)(169,418)(170,420)(171,419)(172,406)(173,408)(174,407)(175,412)
(176,414)(177,413)(178,409)(179,411)(180,410)(181,424)(182,426)(183,425)
(184,430)(185,432)(186,431)(187,427)(188,429)(189,428)(190,446)(191,445)
(192,447)(193,443)(194,442)(195,444)(196,449)(197,448)(198,450)(199,437)
(200,436)(201,438)(202,434)(203,433)(204,435)(205,440)(206,439)(207,441)
(208,455)(209,454)(210,456)(211,452)(212,451)(213,453)(214,458)(215,457)
(216,459)(217,476)(218,475)(219,477)(220,473)(221,472)(222,474)(223,470)
(224,469)(225,471)(226,467)(227,466)(228,468)(229,464)(230,463)(231,465)
(232,461)(233,460)(234,462)(235,485)(236,484)(237,486)(238,482)(239,481)
(240,483)(241,479)(242,478)(243,480);
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*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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