Polytope of Type {3,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1786)
Rank : 3
Schlafli Type : {3,12}
Number of vertices, edges, etc : 54, 324, 216
Order of s0s1s2 : 18
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) :
   3-fold quotients : {3,12}*432
   4-fold quotients : {3,6}*324
   9-fold quotients : {3,12}*144
   12-fold quotients : {3,6}*108
   27-fold quotients : {3,4}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {3,4}*24
   108-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*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      108 facets:
         108 of {3}*6
      36 vertex figures:
         18 of {12}*24
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      108 facets:
         108 of {3}*6
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1> of order 3.
      72 facets:
         72 of {3}*6
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      72 facets:
         72 of {3}*6
      22 vertex figures:
         16 of {12}*24
         6 of {4}*8
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1, s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
      36 facets:
         36 of {3}*6
      16 vertex figures:
         6 of {12}*24
         4 of {6}*12
         6 of {2}*4
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 6.
      36 facets:
         36 of {3}*6
      14 vertex figures:
         3 of {4}*8
         6 of {6}*12
         5 of {12}*24
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2, s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      36 facets:
         36 of {3}*6
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
      36 facets:
         36 of {3}*6
      11 vertex figures:
         8 of {12}*24
         3 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
      36 facets:
         36 of {3}*6
      12 vertex figures:
         6 of {12}*24
         6 of {6}*12

Permutation Representation (GAP) : 
s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 21)( 14, 22)( 15, 24)( 16, 23)( 19, 20)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 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, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)(109,225)(110,226)(111,228)(112,227)(113,221)(114,222)(115,224)(116,223)(117,217)(118,218)(119,220)(120,219)(121,233)(122,234)(123,236)(124,235)(125,229)(126,230)(127,232)(128,231)(129,237)(130,238)(131,240)(132,239)(133,241)(134,242)(135,244)(136,243)(137,249)(138,250)(139,252)(140,251)(141,245)(142,246)(143,248)(144,247)(145,297)(146,298)(147,300)(148,299)(149,293)(150,294)(151,296)(152,295)(153,289)(154,290)(155,292)(156,291)(157,305)(158,306)(159,308)(160,307)(161,301)(162,302)(163,304)(164,303)(165,309)(166,310)(167,312)(168,311)(169,313)(170,314)(171,316)(172,315)(173,321)(174,322)(175,324)(176,323)(177,317)(178,318)(179,320)(180,319)(181,261)(182,262)(183,264)(184,263)(185,257)(186,258)(187,260)(188,259)(189,253)(190,254)(191,256)(192,255)(193,269)(194,270)(195,272)(196,271)(197,265)(198,266)(199,268)(200,267)(201,273)(202,274)(203,276)(204,275)(205,277)(206,278)(207,280)(208,279)(209,285)(210,286)(211,288)(212,287)(213,281)(214,282)(215,284)(216,283);;
s1 := (  1,265)(  2,267)(  3,266)(  4,268)(  5,273)(  6,275)(  7,274)(  8,276)(  9,269)( 10,271)( 11,270)( 12,272)( 13,277)( 14,279)( 15,278)( 16,280)( 17,285)( 18,287)( 19,286)( 20,288)( 21,281)( 22,283)( 23,282)( 24,284)( 25,253)( 26,255)( 27,254)( 28,256)( 29,261)( 30,263)( 31,262)( 32,264)( 33,257)( 34,259)( 35,258)( 36,260)( 37,241)( 38,243)( 39,242)( 40,244)( 41,249)( 42,251)( 43,250)( 44,252)( 45,245)( 46,247)( 47,246)( 48,248)( 49,217)( 50,219)( 51,218)( 52,220)( 53,225)( 54,227)( 55,226)( 56,228)( 57,221)( 58,223)( 59,222)( 60,224)( 61,229)( 62,231)( 63,230)( 64,232)( 65,237)( 66,239)( 67,238)( 68,240)( 69,233)( 70,235)( 71,234)( 72,236)( 73,297)( 74,299)( 75,298)( 76,300)( 77,293)( 78,295)( 79,294)( 80,296)( 81,289)( 82,291)( 83,290)( 84,292)( 85,309)( 86,311)( 87,310)( 88,312)( 89,305)( 90,307)( 91,306)( 92,308)( 93,301)( 94,303)( 95,302)( 96,304)( 97,321)( 98,323)( 99,322)(100,324)(101,317)(102,319)(103,318)(104,320)(105,313)(106,315)(107,314)(108,316)(109,157)(110,159)(111,158)(112,160)(113,165)(114,167)(115,166)(116,168)(117,161)(118,163)(119,162)(120,164)(121,169)(122,171)(123,170)(124,172)(125,177)(126,179)(127,178)(128,180)(129,173)(130,175)(131,174)(132,176)(133,145)(134,147)(135,146)(136,148)(137,153)(138,155)(139,154)(140,156)(141,149)(142,151)(143,150)(144,152)(181,189)(182,191)(183,190)(184,192)(186,187)(193,201)(194,203)(195,202)(196,204)(198,199)(205,213)(206,215)(207,214)(208,216)(210,211);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 30)( 14, 29)( 15, 32)( 16, 31)( 17, 34)( 18, 33)( 19, 36)( 20, 35)( 21, 26)( 22, 25)( 23, 28)( 24, 27)( 37, 74)( 38, 73)( 39, 76)( 40, 75)( 41, 78)( 42, 77)( 43, 80)( 44, 79)( 45, 82)( 46, 81)( 47, 84)( 48, 83)( 49,102)( 50,101)( 51,104)( 52,103)( 53,106)( 54,105)( 55,108)( 56,107)( 57, 98)( 58, 97)( 59,100)( 60, 99)( 61, 94)( 62, 93)( 63, 96)( 64, 95)( 65, 86)( 66, 85)( 67, 88)( 68, 87)( 69, 90)( 70, 89)( 71, 92)( 72, 91)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)(128,143)(129,134)(130,133)(131,136)(132,135)(145,182)(146,181)(147,184)(148,183)(149,186)(150,185)(151,188)(152,187)(153,190)(154,189)(155,192)(156,191)(157,210)(158,209)(159,212)(160,211)(161,214)(162,213)(163,216)(164,215)(165,206)(166,205)(167,208)(168,207)(169,202)(170,201)(171,204)(172,203)(173,194)(174,193)(175,196)(176,195)(177,198)(178,197)(179,200)(180,199)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,246)(230,245)(231,248)(232,247)(233,250)(234,249)(235,252)(236,251)(237,242)(238,241)(239,244)(240,243)(253,290)(254,289)(255,292)(256,291)(257,294)(258,293)(259,296)(260,295)(261,298)(262,297)(263,300)(264,299)(265,318)(266,317)(267,320)(268,319)(269,322)(270,321)(271,324)(272,323)(273,314)(274,313)(275,316)(276,315)(277,310)(278,309)(279,312)(280,311)(281,302)(282,301)(283,304)(284,303)(285,306)(286,305)(287,308)(288,307);;
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, 
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*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(324)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 21)( 14, 22)( 15, 24)( 16, 23)( 19, 20)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 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, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)(109,225)(110,226)(111,228)(112,227)(113,221)(114,222)(115,224)(116,223)(117,217)(118,218)(119,220)(120,219)(121,233)(122,234)(123,236)(124,235)(125,229)(126,230)(127,232)(128,231)(129,237)(130,238)(131,240)(132,239)(133,241)(134,242)(135,244)(136,243)(137,249)(138,250)(139,252)(140,251)(141,245)(142,246)(143,248)(144,247)(145,297)(146,298)(147,300)(148,299)(149,293)(150,294)(151,296)(152,295)(153,289)(154,290)(155,292)(156,291)(157,305)(158,306)(159,308)(160,307)(161,301)(162,302)(163,304)(164,303)(165,309)(166,310)(167,312)(168,311)(169,313)(170,314)(171,316)(172,315)(173,321)(174,322)(175,324)(176,323)(177,317)(178,318)(179,320)(180,319)(181,261)(182,262)(183,264)(184,263)(185,257)(186,258)(187,260)(188,259)(189,253)(190,254)(191,256)(192,255)(193,269)(194,270)(195,272)(196,271)(197,265)(198,266)(199,268)(200,267)(201,273)(202,274)(203,276)(204,275)(205,277)(206,278)(207,280)(208,279)(209,285)(210,286)(211,288)(212,287)(213,281)(214,282)(215,284)(216,283);
s1 := Sym(324)!(  1,265)(  2,267)(  3,266)(  4,268)(  5,273)(  6,275)(  7,274)(  8,276)(  9,269)( 10,271)( 11,270)( 12,272)( 13,277)( 14,279)( 15,278)( 16,280)( 17,285)( 18,287)( 19,286)( 20,288)( 21,281)( 22,283)( 23,282)( 24,284)( 25,253)( 26,255)( 27,254)( 28,256)( 29,261)( 30,263)( 31,262)( 32,264)( 33,257)( 34,259)( 35,258)( 36,260)( 37,241)( 38,243)( 39,242)( 40,244)( 41,249)( 42,251)( 43,250)( 44,252)( 45,245)( 46,247)( 47,246)( 48,248)( 49,217)( 50,219)( 51,218)( 52,220)( 53,225)( 54,227)( 55,226)( 56,228)( 57,221)( 58,223)( 59,222)( 60,224)( 61,229)( 62,231)( 63,230)( 64,232)( 65,237)( 66,239)( 67,238)( 68,240)( 69,233)( 70,235)( 71,234)( 72,236)( 73,297)( 74,299)( 75,298)( 76,300)( 77,293)( 78,295)( 79,294)( 80,296)( 81,289)( 82,291)( 83,290)( 84,292)( 85,309)( 86,311)( 87,310)( 88,312)( 89,305)( 90,307)( 91,306)( 92,308)( 93,301)( 94,303)( 95,302)( 96,304)( 97,321)( 98,323)( 99,322)(100,324)(101,317)(102,319)(103,318)(104,320)(105,313)(106,315)(107,314)(108,316)(109,157)(110,159)(111,158)(112,160)(113,165)(114,167)(115,166)(116,168)(117,161)(118,163)(119,162)(120,164)(121,169)(122,171)(123,170)(124,172)(125,177)(126,179)(127,178)(128,180)(129,173)(130,175)(131,174)(132,176)(133,145)(134,147)(135,146)(136,148)(137,153)(138,155)(139,154)(140,156)(141,149)(142,151)(143,150)(144,152)(181,189)(182,191)(183,190)(184,192)(186,187)(193,201)(194,203)(195,202)(196,204)(198,199)(205,213)(206,215)(207,214)(208,216)(210,211);
s2 := Sym(324)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 30)( 14, 29)( 15, 32)( 16, 31)( 17, 34)( 18, 33)( 19, 36)( 20, 35)( 21, 26)( 22, 25)( 23, 28)( 24, 27)( 37, 74)( 38, 73)( 39, 76)( 40, 75)( 41, 78)( 42, 77)( 43, 80)( 44, 79)( 45, 82)( 46, 81)( 47, 84)( 48, 83)( 49,102)( 50,101)( 51,104)( 52,103)( 53,106)( 54,105)( 55,108)( 56,107)( 57, 98)( 58, 97)( 59,100)( 60, 99)( 61, 94)( 62, 93)( 63, 96)( 64, 95)( 65, 86)( 66, 85)( 67, 88)( 68, 87)( 69, 90)( 70, 89)( 71, 92)( 72, 91)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)(128,143)(129,134)(130,133)(131,136)(132,135)(145,182)(146,181)(147,184)(148,183)(149,186)(150,185)(151,188)(152,187)(153,190)(154,189)(155,192)(156,191)(157,210)(158,209)(159,212)(160,211)(161,214)(162,213)(163,216)(164,215)(165,206)(166,205)(167,208)(168,207)(169,202)(170,201)(171,204)(172,203)(173,194)(174,193)(175,196)(176,195)(177,198)(178,197)(179,200)(180,199)(217,218)(219,220)(221,222)(223,224)(225,226)(227,228)(229,246)(230,245)(231,248)(232,247)(233,250)(234,249)(235,252)(236,251)(237,242)(238,241)(239,244)(240,243)(253,290)(254,289)(255,292)(256,291)(257,294)(258,293)(259,296)(260,295)(261,298)(262,297)(263,300)(264,299)(265,318)(266,317)(267,320)(268,319)(269,322)(270,321)(271,324)(272,323)(273,314)(274,313)(275,316)(276,315)(277,310)(278,309)(279,312)(280,311)(281,302)(282,301)(283,304)(284,303)(285,306)(286,305)(287,308)(288,307);
poly := sub<Sym(324)|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, 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*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle