Polytope of Type {6,54}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*1944e
if this polytope has a name.
Group : SmallGroup(1944,954)
Rank : 3
Schlafli Type : {6,54}
Number of vertices, edges, etc : 18, 486, 162
Order of s0s1s2 : 54
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 : {6,27}*972c
   3-fold quotients : {6,18}*648a
   6-fold quotients : {6,9}*324a
   9-fold quotients : {6,18}*216b, {6,6}*216a
   18-fold quotients : {6,9}*108, {6,3}*108
   27-fold quotients : {2,18}*72, {6,6}*72b
   54-fold quotients : {2,9}*36, {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.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      6 vertex figures:
         6 of {54}*108

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

Twisty Puzzle