Polytope of Type {18,54}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,54}*1944a
Also Known As : {18,54|2}. if this polytope has another name.
Group : SmallGroup(1944,947)
Rank : 3
Schlafli Type : {18,54}
Number of vertices, edges, etc : 18, 486, 54
Order of s0s1s2 : 54
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,18}*648a, {6,54}*648a
   9-fold quotients : {2,54}*216, {6,18}*216a, {18,6}*216a
   18-fold quotients : {2,27}*108
   27-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a
   54-fold quotients : {2,9}*36, {9,2}*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.
Irregular Quotients (of which this is a minimal cover):
   None.

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