Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,18}

Atlas Canonical Name {6,18}*1944e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1944,944)
Rank
3
Schläfli Type
{6,18}
Vertices, edges, …
54, 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

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*(s2*s1)^2*s0*(s1*s2)^2> of order 3

54 facets

18 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

18 vertex figures

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

54 facets

18 vertex figures

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

54 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

6 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)( 28,157)( 29,159)( 30,158)( 31,154)( 32,156)( 33,155)( 34,160)( 35,162)( 36,161)( 37,140)( 38,139)( 39,141)( 40,137)( 41,136)( 42,138)( 43,143)( 44,142)( 45,144)( 46,150)( 47,149)( 48,148)( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,122)( 56,121)( 57,123)( 58,119)( 59,118)( 60,120)( 61,125)( 62,124)( 63,126)( 64,132)( 65,131)( 66,130)( 67,129)( 68,128)( 69,127)( 70,135)( 71,134)( 72,133)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)( 79,115)( 80,117)( 81,116)(164,165)(166,169)(167,171)(168,170)(172,173)(175,179)(176,178)(177,180)(181,183)(184,189)(185,188)(186,187)(190,238)(191,240)(192,239)(193,235)(194,237)(195,236)(196,241)(197,243)(198,242)(199,221)(200,220)(201,222)(202,218)(203,217)(204,219)(205,224)(206,223)(207,225)(208,231)(209,230)(210,229)(211,228)(212,227)(213,226)(214,234)(215,233)(216,232)(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)(271,400)(272,402)(273,401)(274,397)(275,399)(276,398)(277,403)(278,405)(279,404)(280,383)(281,382)(282,384)(283,380)(284,379)(285,381)(286,386)(287,385)(288,387)(289,393)(290,392)(291,391)(292,390)(293,389)(294,388)(295,396)(296,395)(297,394)(298,365)(299,364)(300,366)(301,362)(302,361)(303,363)(304,368)(305,367)(306,369)(307,375)(308,374)(309,373)(310,372)(311,371)(312,370)(313,378)(314,377)(315,376)(316,355)(317,357)(318,356)(319,352)(320,354)(321,353)(322,358)(323,360)(324,359)(407,408)(409,412)(410,414)(411,413)(415,416)(418,422)(419,421)(420,423)(424,426)(427,432)(428,431)(429,430)(433,481)(434,483)(435,482)(436,478)(437,480)(438,479)(439,484)(440,486)(441,485)(442,464)(443,463)(444,465)(445,461)(446,460)(447,462)(448,467)(449,466)(450,468)(451,474)(452,473)(453,472)(454,471)(455,470)(456,469)(457,477)(458,476)(459,475);;
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)( 55,299)( 56,298)( 57,300)( 58,301)( 59,303)( 60,302)( 61,306)( 62,305)( 63,304)( 64,321)( 65,320)( 66,319)( 67,323)( 68,322)( 69,324)( 70,316)( 71,318)( 72,317)( 73,313)( 74,315)( 75,314)( 76,309)( 77,308)( 78,307)( 79,311)( 80,310)( 81,312)( 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)(136,380)(137,379)(138,381)(139,382)(140,384)(141,383)(142,387)(143,386)(144,385)(145,402)(146,401)(147,400)(148,404)(149,403)(150,405)(151,397)(152,399)(153,398)(154,394)(155,396)(156,395)(157,390)(158,389)(159,388)(160,392)(161,391)(162,393)(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)(217,461)(218,460)(219,462)(220,463)(221,465)(222,464)(223,468)(224,467)(225,466)(226,483)(227,482)(228,481)(229,485)(230,484)(231,486)(232,478)(233,480)(234,479)(235,475)(236,477)(237,476)(238,471)(239,470)(240,469)(241,473)(242,472)(243,474);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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)( 28,157)( 29,159)( 30,158)( 31,154)( 32,156)( 33,155)( 34,160)( 35,162)( 36,161)( 37,140)( 38,139)( 39,141)( 40,137)( 41,136)( 42,138)( 43,143)( 44,142)( 45,144)( 46,150)( 47,149)( 48,148)( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,122)( 56,121)( 57,123)( 58,119)( 59,118)( 60,120)( 61,125)( 62,124)( 63,126)( 64,132)( 65,131)( 66,130)( 67,129)( 68,128)( 69,127)( 70,135)( 71,134)( 72,133)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)( 79,115)( 80,117)( 81,116)(164,165)(166,169)(167,171)(168,170)(172,173)(175,179)(176,178)(177,180)(181,183)(184,189)(185,188)(186,187)(190,238)(191,240)(192,239)(193,235)(194,237)(195,236)(196,241)(197,243)(198,242)(199,221)(200,220)(201,222)(202,218)(203,217)(204,219)(205,224)(206,223)(207,225)(208,231)(209,230)(210,229)(211,228)(212,227)(213,226)(214,234)(215,233)(216,232)(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)(271,400)(272,402)(273,401)(274,397)(275,399)(276,398)(277,403)(278,405)(279,404)(280,383)(281,382)(282,384)(283,380)(284,379)(285,381)(286,386)(287,385)(288,387)(289,393)(290,392)(291,391)(292,390)(293,389)(294,388)(295,396)(296,395)(297,394)(298,365)(299,364)(300,366)(301,362)(302,361)(303,363)(304,368)(305,367)(306,369)(307,375)(308,374)(309,373)(310,372)(311,371)(312,370)(313,378)(314,377)(315,376)(316,355)(317,357)(318,356)(319,352)(320,354)(321,353)(322,358)(323,360)(324,359)(407,408)(409,412)(410,414)(411,413)(415,416)(418,422)(419,421)(420,423)(424,426)(427,432)(428,431)(429,430)(433,481)(434,483)(435,482)(436,478)(437,480)(438,479)(439,484)(440,486)(441,485)(442,464)(443,463)(444,465)(445,461)(446,460)(447,462)(448,467)(449,466)(450,468)(451,474)(452,473)(453,472)(454,471)(455,470)(456,469)(457,477)(458,476)(459,475);
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)( 55,299)( 56,298)( 57,300)( 58,301)( 59,303)( 60,302)( 61,306)( 62,305)( 63,304)( 64,321)( 65,320)( 66,319)( 67,323)( 68,322)( 69,324)( 70,316)( 71,318)( 72,317)( 73,313)( 74,315)( 75,314)( 76,309)( 77,308)( 78,307)( 79,311)( 80,310)( 81,312)( 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)(136,380)(137,379)(138,381)(139,382)(140,384)(141,383)(142,387)(143,386)(144,385)(145,402)(146,401)(147,400)(148,404)(149,403)(150,405)(151,397)(152,399)(153,398)(154,394)(155,396)(156,395)(157,390)(158,389)(159,388)(160,392)(161,391)(162,393)(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)(217,461)(218,460)(219,462)(220,463)(221,465)(222,464)(223,468)(224,467)(225,466)(226,483)(227,482)(228,481)(229,485)(230,484)(231,486)(232,478)(233,480)(234,479)(235,475)(236,477)(237,476)(238,471)(239,470)(240,469)(241,473)(242,472)(243,474);
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

Twisty Puzzle