Polytope of Type {18,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1944j
if this polytope has a name.
Group : SmallGroup(1944,952)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 54
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 : {18,6}*972d
   3-fold quotients : {18,6}*648b, {6,6}*648b, {18,6}*648f, {18,6}*648g
   6-fold quotients : {18,6}*324a, {6,6}*324a, {18,6}*324b, {18,6}*324c
   9-fold quotients : {18,6}*216a, {6,6}*216b
   18-fold quotients : {6,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72a
   54-fold quotients : {9,2}*36
   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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      30 facets:
         6 of {9}*18
         24 of {18}*36
      81 vertex figures:
         81 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      66 vertex figures:
         48 of {6}*12
         18 of {2}*4
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
      18 facets:
         18 of {18}*36
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {9}*18
         6 of {18}*36
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {9}*18
         6 of {18}*36
      33 vertex figures:
         24 of {6}*12
         9 of {2}*4

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

Twisty Puzzle