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}*1728a
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 : 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 : {6,6}*864b
   3-fold quotients : {6,12}*576a
   4-fold quotients : {6,12}*432a, {6,3}*432
   6-fold quotients : {6,6}*288a
   8-fold quotients : {6,6}*216a
   9-fold quotients : {6,12}*192a
   12-fold quotients : {6,12}*144b, {6,3}*144
   16-fold quotients : {6,3}*108
   18-fold quotients : {6,6}*96
   24-fold quotients : {6,6}*72b
   36-fold quotients : {2,12}*48, {3,6}*48, {6,3}*48
   48-fold quotients : {6,3}*36
   72-fold quotients : {3,3}*24, {2,6}*24
   108-fold quotients : {2,4}*16
   144-fold quotients : {2,3}*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=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*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*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*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*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
      48 facets:
         48 of {6}*12
      24 vertex figures:
         24 of {12}*24
   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*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*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=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1> of order 6.
      24 facets:
         24 of {6}*12
      12 vertex figures:
         12 of {12}*24
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
      24 facets:
         24 of {6}*12
      12 vertex figures:
         12 of {12}*24

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

Twisty Puzzle