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}*1944d
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 : {9,6}*972b
   3-fold quotients : {18,6}*648d, {18,6}*648e
   6-fold quotients : {9,6}*324b, {9,6}*324d
   9-fold quotients : {6,6}*216c
   18-fold quotients : {3,6}*108
   27-fold quotients : {6,6}*72c
   54-fold quotients : {3,6}*36
   81-fold quotients : {6,2}*24
   162-fold quotients : {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=<s1*s2*s1*s0*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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 3.
      36 facets:
         9 of {18}*36
         27 of {6}*12
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1> of order 3.
      18 facets:
         18 of {18}*36
      66 vertex figures:
         48 of {6}*12
         18 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 9.
      12 facets:
         3 of {18}*36
         9 of {6}*12
      18 vertex figures:
         18 of {6}*12

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

Twisty Puzzle