Polytope of Type {9,24}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,24}*1728
if this polytope has a name.
Group : SmallGroup(1728,12315)
Rank : 3
Schlafli Type : {9,24}
Number of vertices, edges, etc : 36, 432, 96
Order of s0s1s2 : 18
Order of s0s1s2s1 : 24
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) :
   3-fold quotients : {9,8}*576, {3,24}*576
   4-fold quotients : {9,12}*432
   9-fold quotients : {3,8}*192
   12-fold quotients : {9,4}*144, {3,12}*144
   16-fold quotients : {9,6}*108
   24-fold quotients : {9,4}*72
   36-fold quotients : {3,4}*48
   48-fold quotients : {9,2}*36, {3,6}*36
   72-fold quotients : {3,4}*24
   144-fold quotients : {3,2}*12
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*s2*s1*s2*s1*s2> of order 2.
      48 facets:
         48 of {9}*18
      18 vertex figures:
         18 of {24}*48
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      48 facets:
         48 of {9}*18
      24 vertex figures:
         12 of {24}*48
         12 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {9}*18
      12 vertex figures:
         6 of {24}*48
         6 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 4.
      24 facets:
         24 of {9}*18
      18 vertex figures:
         6 of {24}*48
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1, s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         24 of {9}*18
      12 vertex figures:
         6 of {24}*48
         6 of {12}*24

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

Twisty Puzzle