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}*1296h
if this polytope has a name.
Group : SmallGroup(1296,3529)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 54, 324, 54
Order of s0s1s2 : 6
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) :
   3-fold quotients : {12,4}*432b
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72, {6,6}*72c
   36-fold quotients : {3,6}*36
   54-fold quotients : {6,2}*24
   108-fold quotients : {3,2}*12
   162-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*s1*s2*s1*s0*s1*s2*s1> of order 2.
      27 facets:
         27 of {12}*24
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      27 facets:
         27 of {12}*24
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      30 facets:
         6 of {6}*12
         24 of {12}*24
      30 vertex figures:
         24 of {12}*24
         6 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 4.
      15 facets:
         3 of {6}*12
         12 of {12}*24
      15 vertex figures:
         12 of {12}*24
         3 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      12 vertex figures:
         6 of {12}*24
         6 of {6}*12

Permutation Representation (GAP) : 
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);;
s1 := ( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,32)(11,31)(12,33)(13,29)(14,28)(15,30)(16,35)(17,34)(18,36)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(37,41)(38,40)(39,42)(43,44)(46,68)(47,67)(48,69)(49,65)(50,64)(51,66)(52,71)(53,70)(54,72)(73,77)(74,76)(75,78)(79,80);;
s2 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(80,81);;
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);
s1 := Sym(81)!( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,32)(11,31)(12,33)(13,29)(14,28)(15,30)(16,35)(17,34)(18,36)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,62)(26,61)(27,63)(37,41)(38,40)(39,42)(43,44)(46,68)(47,67)(48,69)(49,65)(50,64)(51,66)(52,71)(53,70)(54,72)(73,77)(74,76)(75,78)(79,80);
s2 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)(23,24)(26,27)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)(47,48)(50,51)(53,54)(55,64)(56,66)(57,65)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(74,75)(77,78)(80,81);
poly := sub<Sym(81)|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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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