Polytope of Type {6,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1944l
if this polytope has a name.
Group : SmallGroup(1944,956)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 54, 486, 162
Order of s0s1s2 : 6
Order of s0s1s2s1 : 18
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}*972e
   3-fold quotients : {6,18}*648g
   6-fold quotients : {6,18}*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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      30 vertex figures:
         12 of {18}*36
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 9.
      18 facets:
         18 of {6}*12
      22 vertex figures:
         4 of {18}*36
         18 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 9.
      18 facets:
         18 of {6}*12
      10 vertex figures:
         4 of {18}*36
         6 of {6}*12

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

Twisty Puzzle