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}*1944j
if this polytope has a name.
Group : SmallGroup(1944,945)
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
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,9}*972d
   3-fold quotients : {6,18}*648e, {18,6}*648h
   6-fold quotients : {6,9}*324d, {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 := (  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)( 82,167)( 83,168)( 84,166)( 85,165)( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)( 93,184)( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)(101,177)(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)(109,194)(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)(117,198)(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)(125,215)(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)(133,205)(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)(141,218)(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)(149,235)(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)(157,228)(158,226)(159,227)(160,232)(161,233)(162,234)(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)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)(331,412)(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)(339,425)(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)(347,415)(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)(355,435)(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)(363,454)(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)(371,447)(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)(379,464)(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)(387,468)(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)(395,485)(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)(403,475)(404,476)(405,477);;
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)( 28,158)( 29,157)( 30,159)( 31,155)( 32,154)( 33,156)( 34,161)( 35,160)( 36,162)( 37,141)( 38,140)( 39,139)( 40,138)( 41,137)( 42,136)( 43,144)( 44,143)( 45,142)( 46,148)( 47,150)( 48,149)( 49,145)( 50,147)( 51,146)( 52,151)( 53,153)( 54,152)( 55,123)( 56,122)( 57,121)( 58,120)( 59,119)( 60,118)( 61,126)( 62,125)( 63,124)( 64,130)( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,133)( 71,135)( 72,134)( 73,113)( 74,112)( 75,114)( 76,110)( 77,109)( 78,111)( 79,116)( 80,115)( 81,117)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)(174,175)(178,180)(181,184)(182,186)(183,185)(188,189)(190,243)(191,242)(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,223)(200,225)(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)(208,233)(209,232)(210,234)(211,230)(212,229)(213,231)(214,227)(215,226)(216,228)(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)(271,401)(272,400)(273,402)(274,398)(275,397)(276,399)(277,404)(278,403)(279,405)(280,384)(281,383)(282,382)(283,381)(284,380)(285,379)(286,387)(287,386)(288,385)(289,391)(290,393)(291,392)(292,388)(293,390)(294,389)(295,394)(296,396)(297,395)(298,366)(299,365)(300,364)(301,363)(302,362)(303,361)(304,369)(305,368)(306,367)(307,373)(308,375)(309,374)(310,370)(311,372)(312,371)(313,376)(314,378)(315,377)(316,356)(317,355)(318,357)(319,353)(320,352)(321,354)(322,359)(323,358)(324,360)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)(417,418)(421,423)(424,427)(425,429)(426,428)(431,432)(433,486)(434,485)(435,484)(436,483)(437,482)(438,481)(439,480)(440,479)(441,478)(442,466)(443,468)(444,467)(445,463)(446,465)(447,464)(448,460)(449,462)(450,461)(451,476)(452,475)(453,477)(454,473)(455,472)(456,474)(457,470)(458,469)(459,471);;
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)( 55,300)( 56,299)( 57,298)( 58,306)( 59,305)( 60,304)( 61,303)( 62,302)( 63,301)( 64,314)( 65,313)( 66,315)( 67,311)( 68,310)( 69,312)( 70,308)( 71,307)( 72,309)( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,322)( 80,324)( 81,323)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)( 88,440)( 89,439)( 90,441)( 91,442)( 92,444)( 93,443)( 94,448)( 95,450)( 96,449)( 97,445)( 98,447)( 99,446)(100,459)(101,458)(102,457)(103,456)(104,455)(105,454)(106,453)(107,452)(108,451)(109,410)(110,409)(111,411)(112,407)(113,406)(114,408)(115,413)(116,412)(117,414)(118,415)(119,417)(120,416)(121,421)(122,423)(123,422)(124,418)(125,420)(126,419)(127,432)(128,431)(129,430)(130,429)(131,428)(132,427)(133,426)(134,425)(135,424)(136,463)(137,465)(138,464)(139,460)(140,462)(141,461)(142,466)(143,468)(144,467)(145,471)(146,470)(147,469)(148,477)(149,476)(150,475)(151,474)(152,473)(153,472)(154,485)(155,484)(156,486)(157,482)(158,481)(159,483)(160,479)(161,478)(162,480)(163,356)(164,355)(165,357)(166,353)(167,352)(168,354)(169,359)(170,358)(171,360)(172,361)(173,363)(174,362)(175,367)(176,369)(177,368)(178,364)(179,366)(180,365)(181,378)(182,377)(183,376)(184,375)(185,374)(186,373)(187,372)(188,371)(189,370)(190,329)(191,328)(192,330)(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,334)(200,336)(201,335)(202,340)(203,342)(204,341)(205,337)(206,339)(207,338)(208,351)(209,350)(210,349)(211,348)(212,347)(213,346)(214,345)(215,344)(216,343)(217,382)(218,384)(219,383)(220,379)(221,381)(222,380)(223,385)(224,387)(225,386)(226,390)(227,389)(228,388)(229,396)(230,395)(231,394)(232,393)(233,392)(234,391)(235,404)(236,403)(237,405)(238,401)(239,400)(240,402)(241,398)(242,397)(243,399);;
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*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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)!(  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)( 82,167)( 83,168)( 84,166)( 85,165)( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)( 93,184)( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)(101,177)(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)(109,194)(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)(117,198)(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)(125,215)(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)(133,205)(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)(141,218)(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)(149,235)(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)(157,228)(158,226)(159,227)(160,232)(161,233)(162,234)(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)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)(331,412)(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)(339,425)(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)(347,415)(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)(355,435)(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)(363,454)(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)(371,447)(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)(379,464)(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)(387,468)(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)(395,485)(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)(403,475)(404,476)(405,477);
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)( 28,158)( 29,157)( 30,159)( 31,155)( 32,154)( 33,156)( 34,161)( 35,160)( 36,162)( 37,141)( 38,140)( 39,139)( 40,138)( 41,137)( 42,136)( 43,144)( 44,143)( 45,142)( 46,148)( 47,150)( 48,149)( 49,145)( 50,147)( 51,146)( 52,151)( 53,153)( 54,152)( 55,123)( 56,122)( 57,121)( 58,120)( 59,119)( 60,118)( 61,126)( 62,125)( 63,124)( 64,130)( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,133)( 71,135)( 72,134)( 73,113)( 74,112)( 75,114)( 76,110)( 77,109)( 78,111)( 79,116)( 80,115)( 81,117)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)(174,175)(178,180)(181,184)(182,186)(183,185)(188,189)(190,243)(191,242)(192,241)(193,240)(194,239)(195,238)(196,237)(197,236)(198,235)(199,223)(200,225)(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)(208,233)(209,232)(210,234)(211,230)(212,229)(213,231)(214,227)(215,226)(216,228)(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)(271,401)(272,400)(273,402)(274,398)(275,397)(276,399)(277,404)(278,403)(279,405)(280,384)(281,383)(282,382)(283,381)(284,380)(285,379)(286,387)(287,386)(288,385)(289,391)(290,393)(291,392)(292,388)(293,390)(294,389)(295,394)(296,396)(297,395)(298,366)(299,365)(300,364)(301,363)(302,362)(303,361)(304,369)(305,368)(306,367)(307,373)(308,375)(309,374)(310,370)(311,372)(312,371)(313,376)(314,378)(315,377)(316,356)(317,355)(318,357)(319,353)(320,352)(321,354)(322,359)(323,358)(324,360)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)(417,418)(421,423)(424,427)(425,429)(426,428)(431,432)(433,486)(434,485)(435,484)(436,483)(437,482)(438,481)(439,480)(440,479)(441,478)(442,466)(443,468)(444,467)(445,463)(446,465)(447,464)(448,460)(449,462)(450,461)(451,476)(452,475)(453,477)(454,473)(455,472)(456,474)(457,470)(458,469)(459,471);
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)( 55,300)( 56,299)( 57,298)( 58,306)( 59,305)( 60,304)( 61,303)( 62,302)( 63,301)( 64,314)( 65,313)( 66,315)( 67,311)( 68,310)( 69,312)( 70,308)( 71,307)( 72,309)( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,322)( 80,324)( 81,323)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)( 88,440)( 89,439)( 90,441)( 91,442)( 92,444)( 93,443)( 94,448)( 95,450)( 96,449)( 97,445)( 98,447)( 99,446)(100,459)(101,458)(102,457)(103,456)(104,455)(105,454)(106,453)(107,452)(108,451)(109,410)(110,409)(111,411)(112,407)(113,406)(114,408)(115,413)(116,412)(117,414)(118,415)(119,417)(120,416)(121,421)(122,423)(123,422)(124,418)(125,420)(126,419)(127,432)(128,431)(129,430)(130,429)(131,428)(132,427)(133,426)(134,425)(135,424)(136,463)(137,465)(138,464)(139,460)(140,462)(141,461)(142,466)(143,468)(144,467)(145,471)(146,470)(147,469)(148,477)(149,476)(150,475)(151,474)(152,473)(153,472)(154,485)(155,484)(156,486)(157,482)(158,481)(159,483)(160,479)(161,478)(162,480)(163,356)(164,355)(165,357)(166,353)(167,352)(168,354)(169,359)(170,358)(171,360)(172,361)(173,363)(174,362)(175,367)(176,369)(177,368)(178,364)(179,366)(180,365)(181,378)(182,377)(183,376)(184,375)(185,374)(186,373)(187,372)(188,371)(189,370)(190,329)(191,328)(192,330)(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,334)(200,336)(201,335)(202,340)(203,342)(204,341)(205,337)(206,339)(207,338)(208,351)(209,350)(210,349)(211,348)(212,347)(213,346)(214,345)(215,344)(216,343)(217,382)(218,384)(219,383)(220,379)(221,381)(222,380)(223,385)(224,387)(225,386)(226,390)(227,389)(228,388)(229,396)(230,395)(231,394)(232,393)(233,392)(234,391)(235,404)(236,403)(237,405)(238,401)(239,400)(240,402)(241,398)(242,397)(243,399);
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*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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