Polytope of Type {6,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30243)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 72, 432, 144
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
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 : {6,12}*864b
   3-fold quotients : {6,12}*576b
   4-fold quotients : {6,12}*432b, {6,12}*432d
   6-fold quotients : {6,12}*288a
   8-fold quotients : {6,6}*216b
   9-fold quotients : {6,4}*192b
   12-fold quotients : {6,12}*144a, {6,12}*144d
   16-fold quotients : {6,6}*108
   18-fold quotients : {6,4}*96
   24-fold quotients : {6,6}*72a
   36-fold quotients : {2,12}*48, {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
   108-fold quotients : {2,4}*16
   144-fold quotients : {2,3}*12, {3,2}*12
   216-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*s2*s1*s2*s1> of order 2.
      72 facets:
         72 of {6}*12
      36 vertex figures:
         36 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 2.
      72 facets:
         72 of {6}*12
      48 vertex figures:
         24 of {12}*24
         24 of {6}*12
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      72 facets:
         72 of {6}*12
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 3.
      48 facets:
         48 of {6}*12
      40 vertex figures:
         16 of {12}*24
         24 of {4}*8
   P/N, where N=<s0*s1*s0*s1> of order 3.
      56 facets:
         12 of {2}*4
         44 of {6}*12
      24 vertex figures:
         24 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 4.
      36 facets:
         36 of {6}*12
      30 vertex figures:
         24 of {6}*12
         6 of {12}*24
   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 6.
      24 facets:
         24 of {6}*12
      20 vertex figures:
         8 of {12}*24
         12 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
      24 facets:
         24 of {6}*12
      20 vertex figures:
         8 of {12}*24
         12 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1> of order 6.
      24 facets:
         24 of {6}*12
      20 vertex figures:
         8 of {12}*24
         12 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      24 facets:
         24 of {6}*12
      32 vertex figures:
         8 of {12}*24
         24 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 6.
      24 facets:
         24 of {6}*12
      24 vertex figures:
         8 of {6}*12
         12 of {4}*8
         4 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1> of order 12.
      12 facets:
         12 of {6}*12
      14 vertex figures:
         8 of {6}*12
         6 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 12.
      12 facets:
         12 of {6}*12
      18 vertex figures:
         4 of {6}*12
         2 of {12}*24
         12 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2, s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1> of order 12.
      12 facets:
         12 of {6}*12
      10 vertex figures:
         4 of {12}*24
         6 of {4}*8

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

Twisty Puzzle