Polytope of Type {12,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728o
if this polytope has a name.
Group : SmallGroup(1728,30326)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 6
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 : {12,12}*576j
   4-fold quotients : {6,12}*432d
   6-fold quotients : {6,12}*288a
   8-fold quotients : {6,6}*216b
   9-fold quotients : {12,4}*192c
   12-fold quotients : {6,12}*144d
   16-fold quotients : {6,6}*108
   18-fold quotients : {6,4}*96
   24-fold quotients : {6,6}*72a
   36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
   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.
      36 facets:
         36 of {12}*24
      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.
      36 facets:
         36 of {12}*24
      48 vertex figures:
         24 of {12}*24
         24 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
      36 facets:
         36 of {12}*24
      36 vertex figures:
         36 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      36 facets:
         36 of {12}*24
      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.
      24 facets:
         24 of {12}*24
      40 vertex figures:
         16 of {12}*24
         24 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1> of order 4.
      18 facets:
         18 of {12}*24
      24 vertex figures:
         12 of {12}*24
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1> of order 4.
      18 facets:
         18 of {12}*24
      24 vertex figures:
         12 of {12}*24
         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 6.
      12 facets:
         12 of {12}*24
      20 vertex figures:
         8 of {12}*24
         12 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      24 vertex figures:
         8 of {6}*12
         4 of {12}*24
         12 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {12}*24
      32 vertex figures:
         8 of {12}*24
         24 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1> of order 6.
      12 facets:
         12 of {12}*24
      20 vertex figures:
         8 of {12}*24
         12 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)(217,325)(218,326)(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)(234,346)(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)(242,350)(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)(250,354)(251,356)(252,355)(253,397)(254,398)(255,400)(256,399)(257,405)(258,406)(259,408)(260,407)(261,401)(262,402)(263,404)(264,403)(265,409)(266,410)(267,412)(268,411)(269,417)(270,418)(271,420)(272,419)(273,413)(274,414)(275,416)(276,415)(277,421)(278,422)(279,424)(280,423)(281,429)(282,430)(283,432)(284,431)(285,425)(286,426)(287,428)(288,427)(289,361)(290,362)(291,364)(292,363)(293,369)(294,370)(295,372)(296,371)(297,365)(298,366)(299,368)(300,367)(301,373)(302,374)(303,376)(304,375)(305,381)(306,382)(307,384)(308,383)(309,377)(310,378)(311,380)(312,379)(313,385)(314,386)(315,388)(316,387)(317,393)(318,394)(319,396)(320,395)(321,389)(322,390)(323,392)(324,391);;
s1 := (  1,253)(  2,256)(  3,255)(  4,254)(  5,257)(  6,260)(  7,259)(  8,258)(  9,261)( 10,264)( 11,263)( 12,262)( 13,285)( 14,288)( 15,287)( 16,286)( 17,277)( 18,280)( 19,279)( 20,278)( 21,281)( 22,284)( 23,283)( 24,282)( 25,269)( 26,272)( 27,271)( 28,270)( 29,273)( 30,276)( 31,275)( 32,274)( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)( 40,218)( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)( 48,226)( 49,249)( 50,252)( 51,251)( 52,250)( 53,241)( 54,244)( 55,243)( 56,242)( 57,245)( 58,248)( 59,247)( 60,246)( 61,233)( 62,236)( 63,235)( 64,234)( 65,237)( 66,240)( 67,239)( 68,238)( 69,229)( 70,232)( 71,231)( 72,230)( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)( 80,294)( 81,297)( 82,300)( 83,299)( 84,298)( 85,321)( 86,324)( 87,323)( 88,322)( 89,313)( 90,316)( 91,315)( 92,314)( 93,317)( 94,320)( 95,319)( 96,318)( 97,305)( 98,308)( 99,307)(100,306)(101,309)(102,312)(103,311)(104,310)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)(120,370)(121,393)(122,396)(123,395)(124,394)(125,385)(126,388)(127,387)(128,386)(129,389)(130,392)(131,391)(132,390)(133,377)(134,380)(135,379)(136,378)(137,381)(138,384)(139,383)(140,382)(141,373)(142,376)(143,375)(144,374)(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)(152,330)(153,333)(154,336)(155,335)(156,334)(157,357)(158,360)(159,359)(160,358)(161,349)(162,352)(163,351)(164,350)(165,353)(166,356)(167,355)(168,354)(169,341)(170,344)(171,343)(172,342)(173,345)(174,348)(175,347)(176,346)(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)(184,398)(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)(192,406)(193,429)(194,432)(195,431)(196,430)(197,421)(198,424)(199,423)(200,422)(201,425)(202,428)(203,427)(204,426)(205,413)(206,416)(207,415)(208,414)(209,417)(210,420)(211,419)(212,418)(213,409)(214,412)(215,411)(216,410);;
s2 := (  1, 14)(  2, 13)(  3, 16)(  4, 15)(  5, 22)(  6, 21)(  7, 24)(  8, 23)(  9, 18)( 10, 17)( 11, 20)( 12, 19)( 25, 26)( 27, 28)( 29, 34)( 30, 33)( 31, 36)( 32, 35)( 37, 50)( 38, 49)( 39, 52)( 40, 51)( 41, 58)( 42, 57)( 43, 60)( 44, 59)( 45, 54)( 46, 53)( 47, 56)( 48, 55)( 61, 62)( 63, 64)( 65, 70)( 66, 69)( 67, 72)( 68, 71)( 73, 86)( 74, 85)( 75, 88)( 76, 87)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 81, 90)( 82, 89)( 83, 92)( 84, 91)( 97, 98)( 99,100)(101,106)(102,105)(103,108)(104,107)(109,122)(110,121)(111,124)(112,123)(113,130)(114,129)(115,132)(116,131)(117,126)(118,125)(119,128)(120,127)(133,134)(135,136)(137,142)(138,141)(139,144)(140,143)(145,158)(146,157)(147,160)(148,159)(149,166)(150,165)(151,168)(152,167)(153,162)(154,161)(155,164)(156,163)(169,170)(171,172)(173,178)(174,177)(175,180)(176,179)(181,194)(182,193)(183,196)(184,195)(185,202)(186,201)(187,204)(188,203)(189,198)(190,197)(191,200)(192,199)(205,206)(207,208)(209,214)(210,213)(211,216)(212,215)(217,338)(218,337)(219,340)(220,339)(221,346)(222,345)(223,348)(224,347)(225,342)(226,341)(227,344)(228,343)(229,326)(230,325)(231,328)(232,327)(233,334)(234,333)(235,336)(236,335)(237,330)(238,329)(239,332)(240,331)(241,350)(242,349)(243,352)(244,351)(245,358)(246,357)(247,360)(248,359)(249,354)(250,353)(251,356)(252,355)(253,374)(254,373)(255,376)(256,375)(257,382)(258,381)(259,384)(260,383)(261,378)(262,377)(263,380)(264,379)(265,362)(266,361)(267,364)(268,363)(269,370)(270,369)(271,372)(272,371)(273,366)(274,365)(275,368)(276,367)(277,386)(278,385)(279,388)(280,387)(281,394)(282,393)(283,396)(284,395)(285,390)(286,389)(287,392)(288,391)(289,410)(290,409)(291,412)(292,411)(293,418)(294,417)(295,420)(296,419)(297,414)(298,413)(299,416)(300,415)(301,398)(302,397)(303,400)(304,399)(305,406)(306,405)(307,408)(308,407)(309,402)(310,401)(311,404)(312,403)(313,422)(314,421)(315,424)(316,423)(317,430)(318,429)(319,432)(320,431)(321,426)(322,425)(323,428)(324,427);;
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*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*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 ];;
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)(217,325)(218,326)(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)(226,330)(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)(234,346)(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)(242,350)(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)(250,354)(251,356)(252,355)(253,397)(254,398)(255,400)(256,399)(257,405)(258,406)(259,408)(260,407)(261,401)(262,402)(263,404)(264,403)(265,409)(266,410)(267,412)(268,411)(269,417)(270,418)(271,420)(272,419)(273,413)(274,414)(275,416)(276,415)(277,421)(278,422)(279,424)(280,423)(281,429)(282,430)(283,432)(284,431)(285,425)(286,426)(287,428)(288,427)(289,361)(290,362)(291,364)(292,363)(293,369)(294,370)(295,372)(296,371)(297,365)(298,366)(299,368)(300,367)(301,373)(302,374)(303,376)(304,375)(305,381)(306,382)(307,384)(308,383)(309,377)(310,378)(311,380)(312,379)(313,385)(314,386)(315,388)(316,387)(317,393)(318,394)(319,396)(320,395)(321,389)(322,390)(323,392)(324,391);
s1 := Sym(432)!(  1,253)(  2,256)(  3,255)(  4,254)(  5,257)(  6,260)(  7,259)(  8,258)(  9,261)( 10,264)( 11,263)( 12,262)( 13,285)( 14,288)( 15,287)( 16,286)( 17,277)( 18,280)( 19,279)( 20,278)( 21,281)( 22,284)( 23,283)( 24,282)( 25,269)( 26,272)( 27,271)( 28,270)( 29,273)( 30,276)( 31,275)( 32,274)( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)( 40,218)( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)( 48,226)( 49,249)( 50,252)( 51,251)( 52,250)( 53,241)( 54,244)( 55,243)( 56,242)( 57,245)( 58,248)( 59,247)( 60,246)( 61,233)( 62,236)( 63,235)( 64,234)( 65,237)( 66,240)( 67,239)( 68,238)( 69,229)( 70,232)( 71,231)( 72,230)( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)( 80,294)( 81,297)( 82,300)( 83,299)( 84,298)( 85,321)( 86,324)( 87,323)( 88,322)( 89,313)( 90,316)( 91,315)( 92,314)( 93,317)( 94,320)( 95,319)( 96,318)( 97,305)( 98,308)( 99,307)(100,306)(101,309)(102,312)(103,311)(104,310)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)(120,370)(121,393)(122,396)(123,395)(124,394)(125,385)(126,388)(127,387)(128,386)(129,389)(130,392)(131,391)(132,390)(133,377)(134,380)(135,379)(136,378)(137,381)(138,384)(139,383)(140,382)(141,373)(142,376)(143,375)(144,374)(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)(152,330)(153,333)(154,336)(155,335)(156,334)(157,357)(158,360)(159,359)(160,358)(161,349)(162,352)(163,351)(164,350)(165,353)(166,356)(167,355)(168,354)(169,341)(170,344)(171,343)(172,342)(173,345)(174,348)(175,347)(176,346)(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)(184,398)(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)(192,406)(193,429)(194,432)(195,431)(196,430)(197,421)(198,424)(199,423)(200,422)(201,425)(202,428)(203,427)(204,426)(205,413)(206,416)(207,415)(208,414)(209,417)(210,420)(211,419)(212,418)(213,409)(214,412)(215,411)(216,410);
s2 := Sym(432)!(  1, 14)(  2, 13)(  3, 16)(  4, 15)(  5, 22)(  6, 21)(  7, 24)(  8, 23)(  9, 18)( 10, 17)( 11, 20)( 12, 19)( 25, 26)( 27, 28)( 29, 34)( 30, 33)( 31, 36)( 32, 35)( 37, 50)( 38, 49)( 39, 52)( 40, 51)( 41, 58)( 42, 57)( 43, 60)( 44, 59)( 45, 54)( 46, 53)( 47, 56)( 48, 55)( 61, 62)( 63, 64)( 65, 70)( 66, 69)( 67, 72)( 68, 71)( 73, 86)( 74, 85)( 75, 88)( 76, 87)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 81, 90)( 82, 89)( 83, 92)( 84, 91)( 97, 98)( 99,100)(101,106)(102,105)(103,108)(104,107)(109,122)(110,121)(111,124)(112,123)(113,130)(114,129)(115,132)(116,131)(117,126)(118,125)(119,128)(120,127)(133,134)(135,136)(137,142)(138,141)(139,144)(140,143)(145,158)(146,157)(147,160)(148,159)(149,166)(150,165)(151,168)(152,167)(153,162)(154,161)(155,164)(156,163)(169,170)(171,172)(173,178)(174,177)(175,180)(176,179)(181,194)(182,193)(183,196)(184,195)(185,202)(186,201)(187,204)(188,203)(189,198)(190,197)(191,200)(192,199)(205,206)(207,208)(209,214)(210,213)(211,216)(212,215)(217,338)(218,337)(219,340)(220,339)(221,346)(222,345)(223,348)(224,347)(225,342)(226,341)(227,344)(228,343)(229,326)(230,325)(231,328)(232,327)(233,334)(234,333)(235,336)(236,335)(237,330)(238,329)(239,332)(240,331)(241,350)(242,349)(243,352)(244,351)(245,358)(246,357)(247,360)(248,359)(249,354)(250,353)(251,356)(252,355)(253,374)(254,373)(255,376)(256,375)(257,382)(258,381)(259,384)(260,383)(261,378)(262,377)(263,380)(264,379)(265,362)(266,361)(267,364)(268,363)(269,370)(270,369)(271,372)(272,371)(273,366)(274,365)(275,368)(276,367)(277,386)(278,385)(279,388)(280,387)(281,394)(282,393)(283,396)(284,395)(285,390)(286,389)(287,392)(288,391)(289,410)(290,409)(291,412)(292,411)(293,418)(294,417)(295,420)(296,419)(297,414)(298,413)(299,416)(300,415)(301,398)(302,397)(303,400)(304,399)(305,406)(306,405)(307,408)(308,407)(309,402)(310,401)(311,404)(312,403)(313,422)(314,421)(315,424)(316,423)(317,430)(318,429)(319,432)(320,431)(321,426)(322,425)(323,428)(324,427);
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*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*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 >;
References : None.
to this polytope

Twisty Puzzle