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}*1944e
if this polytope has a name.
Group : SmallGroup(1944,944)
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}*972b
   3-fold quotients : {6,6}*648a, {18,6}*648g
   6-fold quotients : {6,6}*324b, {18,6}*324c
   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,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 56)( 29, 55)( 30, 57)( 31, 58)( 32, 60)( 33, 59)( 34, 63)( 35, 62)( 36, 61)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 81)( 44, 80)( 45, 79)( 46, 65)( 47, 64)( 48, 66)( 49, 67)( 50, 69)( 51, 68)( 52, 72)( 53, 71)( 54, 70)( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,137)(110,136)(111,138)(112,139)(113,141)(114,140)(115,144)(116,143)(117,142)(118,155)(119,154)(120,156)(121,157)(122,159)(123,158)(124,162)(125,161)(126,160)(127,146)(128,145)(129,147)(130,148)(131,150)(132,149)(133,153)(134,152)(135,151)(164,165)(166,168)(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)(179,187)(180,189)(190,218)(191,217)(192,219)(193,220)(194,222)(195,221)(196,225)(197,224)(198,223)(199,236)(200,235)(201,237)(202,238)(203,240)(204,239)(205,243)(206,242)(207,241)(208,227)(209,226)(210,228)(211,229)(212,231)(213,230)(214,234)(215,233)(216,232)(245,246)(247,249)(250,251)(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)(261,270)(271,299)(272,298)(273,300)(274,301)(275,303)(276,302)(277,306)(278,305)(279,304)(280,317)(281,316)(282,318)(283,319)(284,321)(285,320)(286,324)(287,323)(288,322)(289,308)(290,307)(291,309)(292,310)(293,312)(294,311)(295,315)(296,314)(297,313)(326,327)(328,330)(331,332)(334,343)(335,345)(336,344)(337,348)(338,347)(339,346)(340,350)(341,349)(342,351)(352,380)(353,379)(354,381)(355,382)(356,384)(357,383)(358,387)(359,386)(360,385)(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)(367,405)(368,404)(369,403)(370,389)(371,388)(372,390)(373,391)(374,393)(375,392)(376,396)(377,395)(378,394)(407,408)(409,411)(412,413)(415,424)(416,426)(417,425)(418,429)(419,428)(420,427)(421,431)(422,430)(423,432)(433,461)(434,460)(435,462)(436,463)(437,465)(438,464)(439,468)(440,467)(441,466)(442,479)(443,478)(444,480)(445,481)(446,483)(447,482)(448,486)(449,485)(450,484)(451,470)(452,469)(453,471)(454,472)(455,474)(456,473)(457,477)(458,476)(459,475);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)(  9, 32)( 10, 45)( 11, 44)( 12, 43)( 13, 42)( 14, 41)( 15, 40)( 16, 39)( 17, 38)( 18, 37)( 19, 50)( 20, 49)( 21, 51)( 22, 47)( 23, 46)( 24, 48)( 25, 53)( 26, 52)( 27, 54)( 55, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 70)( 65, 72)( 66, 71)( 68, 69)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82,190)( 83,192)( 84,191)( 85,196)( 86,198)( 87,197)( 88,193)( 89,195)( 90,194)( 91,207)( 92,206)( 93,205)( 94,204)( 95,203)( 96,202)( 97,201)( 98,200)( 99,199)(100,212)(101,211)(102,213)(103,209)(104,208)(105,210)(106,215)(107,214)(108,216)(109,163)(110,165)(111,164)(112,169)(113,171)(114,170)(115,166)(116,168)(117,167)(118,180)(119,179)(120,178)(121,177)(122,176)(123,175)(124,174)(125,173)(126,172)(127,185)(128,184)(129,186)(130,182)(131,181)(132,183)(133,188)(134,187)(135,189)(136,218)(137,217)(138,219)(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,232)(146,234)(147,233)(148,229)(149,231)(150,230)(151,226)(152,228)(153,227)(154,240)(155,239)(156,238)(157,237)(158,236)(159,235)(160,243)(161,242)(162,241)(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,288)(254,287)(255,286)(256,285)(257,284)(258,283)(259,282)(260,281)(261,280)(262,293)(263,292)(264,294)(265,290)(266,289)(267,291)(268,296)(269,295)(270,297)(298,299)(301,305)(302,304)(303,306)(307,313)(308,315)(309,314)(311,312)(316,321)(317,320)(318,319)(322,324)(325,433)(326,435)(327,434)(328,439)(329,441)(330,440)(331,436)(332,438)(333,437)(334,450)(335,449)(336,448)(337,447)(338,446)(339,445)(340,444)(341,443)(342,442)(343,455)(344,454)(345,456)(346,452)(347,451)(348,453)(349,458)(350,457)(351,459)(352,406)(353,408)(354,407)(355,412)(356,414)(357,413)(358,409)(359,411)(360,410)(361,423)(362,422)(363,421)(364,420)(365,419)(366,418)(367,417)(368,416)(369,415)(370,428)(371,427)(372,429)(373,425)(374,424)(375,426)(376,431)(377,430)(378,432)(379,461)(380,460)(381,462)(382,467)(383,466)(384,468)(385,464)(386,463)(387,465)(388,475)(389,477)(390,476)(391,472)(392,474)(393,473)(394,469)(395,471)(396,470)(397,483)(398,482)(399,481)(400,480)(401,479)(402,478)(403,486)(404,485)(405,484);;
s2 := (  1,325)(  2,326)(  3,327)(  4,332)(  5,333)(  6,331)(  7,330)(  8,328)(  9,329)( 10,345)( 11,343)( 12,344)( 13,349)( 14,350)( 15,351)( 16,347)( 17,348)( 18,346)( 19,335)( 20,336)( 21,334)( 22,342)( 23,340)( 24,341)( 25,337)( 26,338)( 27,339)( 28,364)( 29,365)( 30,366)( 31,362)( 32,363)( 33,361)( 34,369)( 35,367)( 36,368)( 37,357)( 38,355)( 39,356)( 40,352)( 41,353)( 42,354)( 43,359)( 44,360)( 45,358)( 46,374)( 47,375)( 48,373)( 49,372)( 50,370)( 51,371)( 52,376)( 53,377)( 54,378)( 55,402)( 56,400)( 57,401)( 58,397)( 59,398)( 60,399)( 61,404)( 62,405)( 63,403)( 64,392)( 65,393)( 66,391)( 67,390)( 68,388)( 69,389)( 70,394)( 71,395)( 72,396)( 73,382)( 74,383)( 75,384)( 76,380)( 77,381)( 78,379)( 79,387)( 80,385)( 81,386)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)( 88,249)( 89,247)( 90,248)( 91,264)( 92,262)( 93,263)( 94,268)( 95,269)( 96,270)( 97,266)( 98,267)( 99,265)(100,254)(101,255)(102,253)(103,261)(104,259)(105,260)(106,256)(107,257)(108,258)(109,283)(110,284)(111,285)(112,281)(113,282)(114,280)(115,288)(116,286)(117,287)(118,276)(119,274)(120,275)(121,271)(122,272)(123,273)(124,278)(125,279)(126,277)(127,293)(128,294)(129,292)(130,291)(131,289)(132,290)(133,295)(134,296)(135,297)(136,321)(137,319)(138,320)(139,316)(140,317)(141,318)(142,323)(143,324)(144,322)(145,311)(146,312)(147,310)(148,309)(149,307)(150,308)(151,313)(152,314)(153,315)(154,301)(155,302)(156,303)(157,299)(158,300)(159,298)(160,306)(161,304)(162,305)(163,406)(164,407)(165,408)(166,413)(167,414)(168,412)(169,411)(170,409)(171,410)(172,426)(173,424)(174,425)(175,430)(176,431)(177,432)(178,428)(179,429)(180,427)(181,416)(182,417)(183,415)(184,423)(185,421)(186,422)(187,418)(188,419)(189,420)(190,445)(191,446)(192,447)(193,443)(194,444)(195,442)(196,450)(197,448)(198,449)(199,438)(200,436)(201,437)(202,433)(203,434)(204,435)(205,440)(206,441)(207,439)(208,455)(209,456)(210,454)(211,453)(212,451)(213,452)(214,457)(215,458)(216,459)(217,483)(218,481)(219,482)(220,478)(221,479)(222,480)(223,485)(224,486)(225,484)(226,473)(227,474)(228,472)(229,471)(230,469)(231,470)(232,475)(233,476)(234,477)(235,463)(236,464)(237,465)(238,461)(239,462)(240,460)(241,468)(242,466)(243,467);;
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*s0*s1*s2*s0*s1*s0*s1*s2*s0*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(486)!(  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 56)( 29, 55)( 30, 57)( 31, 58)( 32, 60)( 33, 59)( 34, 63)( 35, 62)( 36, 61)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 81)( 44, 80)( 45, 79)( 46, 65)( 47, 64)( 48, 66)( 49, 67)( 50, 69)( 51, 68)( 52, 72)( 53, 71)( 54, 70)( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,137)(110,136)(111,138)(112,139)(113,141)(114,140)(115,144)(116,143)(117,142)(118,155)(119,154)(120,156)(121,157)(122,159)(123,158)(124,162)(125,161)(126,160)(127,146)(128,145)(129,147)(130,148)(131,150)(132,149)(133,153)(134,152)(135,151)(164,165)(166,168)(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)(179,187)(180,189)(190,218)(191,217)(192,219)(193,220)(194,222)(195,221)(196,225)(197,224)(198,223)(199,236)(200,235)(201,237)(202,238)(203,240)(204,239)(205,243)(206,242)(207,241)(208,227)(209,226)(210,228)(211,229)(212,231)(213,230)(214,234)(215,233)(216,232)(245,246)(247,249)(250,251)(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)(261,270)(271,299)(272,298)(273,300)(274,301)(275,303)(276,302)(277,306)(278,305)(279,304)(280,317)(281,316)(282,318)(283,319)(284,321)(285,320)(286,324)(287,323)(288,322)(289,308)(290,307)(291,309)(292,310)(293,312)(294,311)(295,315)(296,314)(297,313)(326,327)(328,330)(331,332)(334,343)(335,345)(336,344)(337,348)(338,347)(339,346)(340,350)(341,349)(342,351)(352,380)(353,379)(354,381)(355,382)(356,384)(357,383)(358,387)(359,386)(360,385)(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)(367,405)(368,404)(369,403)(370,389)(371,388)(372,390)(373,391)(374,393)(375,392)(376,396)(377,395)(378,394)(407,408)(409,411)(412,413)(415,424)(416,426)(417,425)(418,429)(419,428)(420,427)(421,431)(422,430)(423,432)(433,461)(434,460)(435,462)(436,463)(437,465)(438,464)(439,468)(440,467)(441,466)(442,479)(443,478)(444,480)(445,481)(446,483)(447,482)(448,486)(449,485)(450,484)(451,470)(452,469)(453,471)(454,472)(455,474)(456,473)(457,477)(458,476)(459,475);
s1 := Sym(486)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)(  9, 32)( 10, 45)( 11, 44)( 12, 43)( 13, 42)( 14, 41)( 15, 40)( 16, 39)( 17, 38)( 18, 37)( 19, 50)( 20, 49)( 21, 51)( 22, 47)( 23, 46)( 24, 48)( 25, 53)( 26, 52)( 27, 54)( 55, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 70)( 65, 72)( 66, 71)( 68, 69)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82,190)( 83,192)( 84,191)( 85,196)( 86,198)( 87,197)( 88,193)( 89,195)( 90,194)( 91,207)( 92,206)( 93,205)( 94,204)( 95,203)( 96,202)( 97,201)( 98,200)( 99,199)(100,212)(101,211)(102,213)(103,209)(104,208)(105,210)(106,215)(107,214)(108,216)(109,163)(110,165)(111,164)(112,169)(113,171)(114,170)(115,166)(116,168)(117,167)(118,180)(119,179)(120,178)(121,177)(122,176)(123,175)(124,174)(125,173)(126,172)(127,185)(128,184)(129,186)(130,182)(131,181)(132,183)(133,188)(134,187)(135,189)(136,218)(137,217)(138,219)(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,232)(146,234)(147,233)(148,229)(149,231)(150,230)(151,226)(152,228)(153,227)(154,240)(155,239)(156,238)(157,237)(158,236)(159,235)(160,243)(161,242)(162,241)(244,271)(245,273)(246,272)(247,277)(248,279)(249,278)(250,274)(251,276)(252,275)(253,288)(254,287)(255,286)(256,285)(257,284)(258,283)(259,282)(260,281)(261,280)(262,293)(263,292)(264,294)(265,290)(266,289)(267,291)(268,296)(269,295)(270,297)(298,299)(301,305)(302,304)(303,306)(307,313)(308,315)(309,314)(311,312)(316,321)(317,320)(318,319)(322,324)(325,433)(326,435)(327,434)(328,439)(329,441)(330,440)(331,436)(332,438)(333,437)(334,450)(335,449)(336,448)(337,447)(338,446)(339,445)(340,444)(341,443)(342,442)(343,455)(344,454)(345,456)(346,452)(347,451)(348,453)(349,458)(350,457)(351,459)(352,406)(353,408)(354,407)(355,412)(356,414)(357,413)(358,409)(359,411)(360,410)(361,423)(362,422)(363,421)(364,420)(365,419)(366,418)(367,417)(368,416)(369,415)(370,428)(371,427)(372,429)(373,425)(374,424)(375,426)(376,431)(377,430)(378,432)(379,461)(380,460)(381,462)(382,467)(383,466)(384,468)(385,464)(386,463)(387,465)(388,475)(389,477)(390,476)(391,472)(392,474)(393,473)(394,469)(395,471)(396,470)(397,483)(398,482)(399,481)(400,480)(401,479)(402,478)(403,486)(404,485)(405,484);
s2 := Sym(486)!(  1,325)(  2,326)(  3,327)(  4,332)(  5,333)(  6,331)(  7,330)(  8,328)(  9,329)( 10,345)( 11,343)( 12,344)( 13,349)( 14,350)( 15,351)( 16,347)( 17,348)( 18,346)( 19,335)( 20,336)( 21,334)( 22,342)( 23,340)( 24,341)( 25,337)( 26,338)( 27,339)( 28,364)( 29,365)( 30,366)( 31,362)( 32,363)( 33,361)( 34,369)( 35,367)( 36,368)( 37,357)( 38,355)( 39,356)( 40,352)( 41,353)( 42,354)( 43,359)( 44,360)( 45,358)( 46,374)( 47,375)( 48,373)( 49,372)( 50,370)( 51,371)( 52,376)( 53,377)( 54,378)( 55,402)( 56,400)( 57,401)( 58,397)( 59,398)( 60,399)( 61,404)( 62,405)( 63,403)( 64,392)( 65,393)( 66,391)( 67,390)( 68,388)( 69,389)( 70,394)( 71,395)( 72,396)( 73,382)( 74,383)( 75,384)( 76,380)( 77,381)( 78,379)( 79,387)( 80,385)( 81,386)( 82,244)( 83,245)( 84,246)( 85,251)( 86,252)( 87,250)( 88,249)( 89,247)( 90,248)( 91,264)( 92,262)( 93,263)( 94,268)( 95,269)( 96,270)( 97,266)( 98,267)( 99,265)(100,254)(101,255)(102,253)(103,261)(104,259)(105,260)(106,256)(107,257)(108,258)(109,283)(110,284)(111,285)(112,281)(113,282)(114,280)(115,288)(116,286)(117,287)(118,276)(119,274)(120,275)(121,271)(122,272)(123,273)(124,278)(125,279)(126,277)(127,293)(128,294)(129,292)(130,291)(131,289)(132,290)(133,295)(134,296)(135,297)(136,321)(137,319)(138,320)(139,316)(140,317)(141,318)(142,323)(143,324)(144,322)(145,311)(146,312)(147,310)(148,309)(149,307)(150,308)(151,313)(152,314)(153,315)(154,301)(155,302)(156,303)(157,299)(158,300)(159,298)(160,306)(161,304)(162,305)(163,406)(164,407)(165,408)(166,413)(167,414)(168,412)(169,411)(170,409)(171,410)(172,426)(173,424)(174,425)(175,430)(176,431)(177,432)(178,428)(179,429)(180,427)(181,416)(182,417)(183,415)(184,423)(185,421)(186,422)(187,418)(188,419)(189,420)(190,445)(191,446)(192,447)(193,443)(194,444)(195,442)(196,450)(197,448)(198,449)(199,438)(200,436)(201,437)(202,433)(203,434)(204,435)(205,440)(206,441)(207,439)(208,455)(209,456)(210,454)(211,453)(212,451)(213,452)(214,457)(215,458)(216,459)(217,483)(218,481)(219,482)(220,478)(221,479)(222,480)(223,485)(224,486)(225,484)(226,473)(227,474)(228,472)(229,471)(230,469)(231,470)(232,475)(233,476)(234,477)(235,463)(236,464)(237,465)(238,461)(239,462)(240,460)(241,468)(242,466)(243,467);
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*s0*s1*s2*s0*s1*s0*s1*s2*s0*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 >;
References : None.
to this polytope

Twisty Puzzle