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}*1944f
if this polytope has a name.
Group : SmallGroup(1944,944)
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 : {6,9}*972c
   3-fold quotients : {6,18}*648d, {6,6}*648c
   6-fold quotients : {6,9}*324b, {6,3}*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*s2*s1*s0*s2*s1*s0*s1*s0*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*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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*s0*s1> of order 3.
      66 facets:
         18 of {2}*4
         48 of {6}*12
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1> of order 9.
      18 facets:
         18 of {6}*12
      6 vertex figures:
         6 of {18}*36

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

Twisty Puzzle