Polytope of Type {24,9}

Play with this polytope as a twisty puzzle

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

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

Twisty Puzzle