Polytope of Type {18,18}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944g
if this polytope has a name.
Group : SmallGroup(1944,944)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 18
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 : {18,9}*972c
   3-fold quotients : {6,18}*648c, {18,6}*648h
   6-fold quotients : {6,9}*324c, {18,3}*324
   9-fold quotients : {6,6}*216a
   18-fold quotients : {6,3}*108
   27-fold quotients : {6,6}*72b
   54-fold quotients : {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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 3.
      18 facets:
         18 of {18}*36
      18 vertex figures:
         18 of {18}*36

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

Twisty Puzzle