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}*1944f
if this polytope has a name.
Group : SmallGroup(1944,954)
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 : {54,6}*972c
   3-fold quotients : {18,6}*648b
   6-fold quotients : {18,6}*324a
   9-fold quotients : {18,6}*216a, {6,6}*216b
   18-fold quotients : {6,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72a
   54-fold quotients : {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):
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      6 facets:
         6 of {54}*108
      54 vertex figures:
         54 of {6}*12

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

Twisty Puzzle