Polytope of Type {6,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*1944i
if this polytope has a name.
Group : SmallGroup(1944,952)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 54, 486, 162
Order of s0s1s2 : 18
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 : {6,9}*972e
   3-fold quotients : {6,18}*648a, {6,18}*648c, {6,18}*648e, {6,6}*648c
   6-fold quotients : {6,9}*324a, {6,9}*324c, {6,9}*324d, {6,3}*324
   9-fold quotients : {6,18}*216b, {6,6}*216a
   18-fold quotients : {6,9}*108, {6,3}*108
   27-fold quotients : {2,18}*72, {6,6}*72b
   54-fold quotients : {2,9}*36, {6,3}*36
   81-fold quotients : {2,6}*24
   162-fold quotients : {2,3}*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> of order 3.
      66 facets:
         18 of {2}*4
         48 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36

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

Twisty Puzzle