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}*1944b
if this polytope has a name.
Group : SmallGroup(1944,940)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
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}*972a
   3-fold quotients : {18,18}*648b, {6,18}*648a
   6-fold quotients : {18,9}*324, {6,9}*324a
   9-fold quotients : {6,18}*216b, {6,6}*216a
   18-fold quotients : {6,9}*108, {6,3}*108
   27-fold quotients : {2,18}*72, {6,6}*72b
   54-fold quotients : {2,9}*36, {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*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {18}*36
      18 vertex figures:
         18 of {18}*36
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {18}*36
      18 vertex figures:
         18 of {18}*36

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

Twisty Puzzle