Polytope of Type {18,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1944c
if this polytope has a name.
Group : SmallGroup(1944,943)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 54
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 : {18,6}*972a
   3-fold quotients : {6,6}*648a, {18,6}*648f
   6-fold quotients : {6,6}*324b, {18,6}*324b
   9-fold quotients : {6,6}*216b
   18-fold quotients : {6,6}*108
   27-fold quotients : {6,6}*72a
   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.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 9.
      6 facets:
         6 of {18}*36
      18 vertex figures:
         18 of {6}*12

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

Twisty Puzzle