Polytope of Type {54,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,948)
Rank : 3
Schlafli Type : {54,6}
Number of vertices, edges, etc : 162, 486, 18
Order of s0s1s2 : 54
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 : {27,6}*972a
   3-fold quotients : {18,6}*648a, {54,6}*648b
   6-fold quotients : {9,6}*324a, {27,6}*324
   9-fold quotients : {54,2}*216, {18,6}*216b, {6,6}*216c
   18-fold quotients : {27,2}*108, {9,6}*108, {3,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72c
   54-fold quotients : {9,2}*36, {3,6}*36
   81-fold quotients : {6,2}*24
   162-fold quotients : {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=<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*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      6 facets:
         6 of {54}*108
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      6 facets:
         6 of {54}*108
      90 vertex figures:
         36 of {6}*12
         54 of {2}*4

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

Twisty Puzzle