Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*1944a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1944,941)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
162, 486, 162
Order of s0s1s2
18
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

18-fold

27-fold

81-fold

162-fold

243-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^3> of order 2

84 facets

81 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^3> of order 2

81 facets

84 vertex figures

P/N, where N=<s0*s2*s1*s0*(s1*s2)^2*(s1*s0)^2*s2*s1> of order 3

54 facets

54 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^2*(s1*s0)^2*s2> of order 3

54 facets

54 vertex figures

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 3

54 facets

54 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2*s1*s0*s1> of order 3

54 facets

54 vertex figures

P/N, where N=<(s0*s1)^2> of order 3

66 facets

54 vertex figures

P/N, where N=<(s1*s2*s1*s0)^2*(s1*s2)^2> of order 3

54 facets

54 vertex figures

P/N, where N=<s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1*s2> of order 3

54 facets

66 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2> of order 3

54 facets

54 vertex figures

P/N, where N=<(s0*s1)^3, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 6

30 facets

27 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 6

28 facets

27 vertex figures

P/N, where N=<(s1*s2*s1*s0)^2*(s1*s2)^2, (s0*s1)^2*(s2*(s1*s0)^2)^2*s2*s1> of order 6

27 facets

28 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2> of order 6

30 facets

27 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1*s2> of order 6

30 facets

33 vertex figures

P/N, where N=<(s0*s1)^3, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 6

30 facets

27 vertex figures

P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^2*s2*(s1*s0)^2*(s1*s2)^2*s1> of order 6

27 facets

30 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0, s0*s2*s1*s0*(s1*s2)^2*(s1*s0)^2*s2*s1> of order 6

27 facets

30 vertex figures

P/N, where N=<(s0*s1)^2, (s1*s2*(s1*s0)^2)^2*s1*s2> of order 6

33 facets

30 vertex figures

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2, s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 6

27 facets

30 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*s2*s1*s0*(s1*s2)^2*(s1*s0)^2*s2> of order 9

22 facets

18 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s2, s0*(s2*s1)^2*s0*(s1*s2)^2> of order 9

18 facets

18 vertex figures

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 9

18 facets

22 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle