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}*1944c
if this polytope has a name.
Group : SmallGroup(1944,943)
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 : {6,18}*972a
   3-fold quotients : {6,6}*648b, {6,18}*648f
   6-fold quotients : {6,6}*324a, {6,18}*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.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*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*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2> of order 3.
      54 facets:
         54 of {6}*12
      18 vertex figures:
         18 of {18}*36
   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.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36

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

Twisty Puzzle