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}*1944d
if this polytope has a name.
Group : SmallGroup(1944,942)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
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 : {18,18}*972b
   3-fold quotients : {6,18}*648g, {18,6}*648g
   6-fold quotients : {6,18}*324c, {18,6}*324c
   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.
      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)( 28,157)( 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)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)(174,175)(178,180)(181,184)(182,186)(183,185)(188,189)(190,242)(191,241)(192,243)(193,239)(194,238)(195,240)(196,236)(197,235)(198,237)(199,225)(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,232)(209,234)(210,233)(211,229)(212,231)(213,230)(214,226)(215,228)(216,227)(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)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)(417,418)(421,423)(424,427)(425,429)(426,428)(431,432)(433,485)(434,484)(435,486)(436,482)(437,481)(438,483)(439,479)(440,478)(441,480)(442,468)(443,467)(444,466)(445,465)(446,464)(447,463)(448,462)(449,461)(450,460)(451,475)(452,477)(453,476)(454,472)(455,474)(456,473)(457,469)(458,471)(459,470);;
s2 := (  1,271)(  2,273)(  3,272)(  4,276)(  5,275)(  6,274)(  7,278)(  8,277)(  9,279)( 10,293)( 11,292)( 12,294)( 13,295)( 14,297)( 15,296)( 16,291)( 17,290)( 18,289)( 19,288)( 20,287)( 21,286)( 22,281)( 23,280)( 24,282)( 25,283)( 26,285)( 27,284)( 28,244)( 29,246)( 30,245)( 31,249)( 32,248)( 33,247)( 34,251)( 35,250)( 36,252)( 37,266)( 38,265)( 39,267)( 40,268)( 41,270)( 42,269)( 43,264)( 44,263)( 45,262)( 46,261)( 47,260)( 48,259)( 49,254)( 50,253)( 51,255)( 52,256)( 53,258)( 54,257)( 55,299)( 56,298)( 57,300)( 58,301)( 59,303)( 60,302)( 61,306)( 62,305)( 63,304)( 64,321)( 65,320)( 66,319)( 67,323)( 68,322)( 69,324)( 70,316)( 71,318)( 72,317)( 73,313)( 74,315)( 75,314)( 76,309)( 77,308)( 78,307)( 79,311)( 80,310)( 81,312)( 82,352)( 83,354)( 84,353)( 85,357)( 86,356)( 87,355)( 88,359)( 89,358)( 90,360)( 91,374)( 92,373)( 93,375)( 94,376)( 95,378)( 96,377)( 97,372)( 98,371)( 99,370)(100,369)(101,368)(102,367)(103,362)(104,361)(105,363)(106,364)(107,366)(108,365)(109,325)(110,327)(111,326)(112,330)(113,329)(114,328)(115,332)(116,331)(117,333)(118,347)(119,346)(120,348)(121,349)(122,351)(123,350)(124,345)(125,344)(126,343)(127,342)(128,341)(129,340)(130,335)(131,334)(132,336)(133,337)(134,339)(135,338)(136,380)(137,379)(138,381)(139,382)(140,384)(141,383)(142,387)(143,386)(144,385)(145,402)(146,401)(147,400)(148,404)(149,403)(150,405)(151,397)(152,399)(153,398)(154,394)(155,396)(156,395)(157,390)(158,389)(159,388)(160,392)(161,391)(162,393)(163,433)(164,435)(165,434)(166,438)(167,437)(168,436)(169,440)(170,439)(171,441)(172,455)(173,454)(174,456)(175,457)(176,459)(177,458)(178,453)(179,452)(180,451)(181,450)(182,449)(183,448)(184,443)(185,442)(186,444)(187,445)(188,447)(189,446)(190,406)(191,408)(192,407)(193,411)(194,410)(195,409)(196,413)(197,412)(198,414)(199,428)(200,427)(201,429)(202,430)(203,432)(204,431)(205,426)(206,425)(207,424)(208,423)(209,422)(210,421)(211,416)(212,415)(213,417)(214,418)(215,420)(216,419)(217,461)(218,460)(219,462)(220,463)(221,465)(222,464)(223,468)(224,467)(225,466)(226,483)(227,482)(228,481)(229,485)(230,484)(231,486)(232,478)(233,480)(234,479)(235,475)(236,477)(237,476)(238,471)(239,470)(240,469)(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*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*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 ];;
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)( 28,157)( 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)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)(174,175)(178,180)(181,184)(182,186)(183,185)(188,189)(190,242)(191,241)(192,243)(193,239)(194,238)(195,240)(196,236)(197,235)(198,237)(199,225)(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,232)(209,234)(210,233)(211,229)(212,231)(213,230)(214,226)(215,228)(216,227)(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)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)(417,418)(421,423)(424,427)(425,429)(426,428)(431,432)(433,485)(434,484)(435,486)(436,482)(437,481)(438,483)(439,479)(440,478)(441,480)(442,468)(443,467)(444,466)(445,465)(446,464)(447,463)(448,462)(449,461)(450,460)(451,475)(452,477)(453,476)(454,472)(455,474)(456,473)(457,469)(458,471)(459,470);
s2 := Sym(486)!(  1,271)(  2,273)(  3,272)(  4,276)(  5,275)(  6,274)(  7,278)(  8,277)(  9,279)( 10,293)( 11,292)( 12,294)( 13,295)( 14,297)( 15,296)( 16,291)( 17,290)( 18,289)( 19,288)( 20,287)( 21,286)( 22,281)( 23,280)( 24,282)( 25,283)( 26,285)( 27,284)( 28,244)( 29,246)( 30,245)( 31,249)( 32,248)( 33,247)( 34,251)( 35,250)( 36,252)( 37,266)( 38,265)( 39,267)( 40,268)( 41,270)( 42,269)( 43,264)( 44,263)( 45,262)( 46,261)( 47,260)( 48,259)( 49,254)( 50,253)( 51,255)( 52,256)( 53,258)( 54,257)( 55,299)( 56,298)( 57,300)( 58,301)( 59,303)( 60,302)( 61,306)( 62,305)( 63,304)( 64,321)( 65,320)( 66,319)( 67,323)( 68,322)( 69,324)( 70,316)( 71,318)( 72,317)( 73,313)( 74,315)( 75,314)( 76,309)( 77,308)( 78,307)( 79,311)( 80,310)( 81,312)( 82,352)( 83,354)( 84,353)( 85,357)( 86,356)( 87,355)( 88,359)( 89,358)( 90,360)( 91,374)( 92,373)( 93,375)( 94,376)( 95,378)( 96,377)( 97,372)( 98,371)( 99,370)(100,369)(101,368)(102,367)(103,362)(104,361)(105,363)(106,364)(107,366)(108,365)(109,325)(110,327)(111,326)(112,330)(113,329)(114,328)(115,332)(116,331)(117,333)(118,347)(119,346)(120,348)(121,349)(122,351)(123,350)(124,345)(125,344)(126,343)(127,342)(128,341)(129,340)(130,335)(131,334)(132,336)(133,337)(134,339)(135,338)(136,380)(137,379)(138,381)(139,382)(140,384)(141,383)(142,387)(143,386)(144,385)(145,402)(146,401)(147,400)(148,404)(149,403)(150,405)(151,397)(152,399)(153,398)(154,394)(155,396)(156,395)(157,390)(158,389)(159,388)(160,392)(161,391)(162,393)(163,433)(164,435)(165,434)(166,438)(167,437)(168,436)(169,440)(170,439)(171,441)(172,455)(173,454)(174,456)(175,457)(176,459)(177,458)(178,453)(179,452)(180,451)(181,450)(182,449)(183,448)(184,443)(185,442)(186,444)(187,445)(188,447)(189,446)(190,406)(191,408)(192,407)(193,411)(194,410)(195,409)(196,413)(197,412)(198,414)(199,428)(200,427)(201,429)(202,430)(203,432)(204,431)(205,426)(206,425)(207,424)(208,423)(209,422)(210,421)(211,416)(212,415)(213,417)(214,418)(215,420)(216,419)(217,461)(218,460)(219,462)(220,463)(221,465)(222,464)(223,468)(224,467)(225,466)(226,483)(227,482)(228,481)(229,485)(230,484)(231,486)(232,478)(233,480)(234,479)(235,475)(236,477)(237,476)(238,471)(239,470)(240,469)(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*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*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 >;
References : None.
to this polytope

Twisty Puzzle