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}*1296c
if this polytope has a name.
Group : SmallGroup(1296,2909)
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 : {4,12}*432a, {12,4}*432b
   6-fold quotients : {4,12}*216
   9-fold quotients : {4,4}*144
   18-fold quotients : {4,4}*72
   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*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2> 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*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*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*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> 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*s1*s2*s1*s0*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)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,55)(20,57)(21,56)(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(37,38)(40,44)(41,43)(42,45)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(73,74)(76,80)(77,79)(78,81);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)(36,62)(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)(47,78)(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);;
s2 := ( 1,39)( 2,38)( 3,37)( 4,42)( 5,41)( 6,40)( 7,45)( 8,44)( 9,43)(10,12)(13,15)(16,18)(19,66)(20,65)(21,64)(22,69)(23,68)(24,67)(25,72)(26,71)(27,70)(28,29)(31,32)(34,35)(46,55)(47,57)(48,56)(49,58)(50,60)(51,59)(52,61)(53,63)(54,62)(73,74)(76,77)(79,80);;
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
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*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)(10,28)(11,30)(12,29)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,55)(20,57)(21,56)(22,61)(23,63)(24,62)(25,58)(26,60)(27,59)(37,38)(40,44)(41,43)(42,45)(46,66)(47,65)(48,64)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(73,74)(76,80)(77,79)(78,81);
s1 := Sym(81)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,58)(29,60)(30,59)(31,55)(32,57)(33,56)(34,61)(35,63)(36,62)(37,67)(38,69)(39,68)(40,64)(41,66)(42,65)(43,70)(44,72)(45,71)(46,76)(47,78)(48,77)(49,73)(50,75)(51,74)(52,79)(53,81)(54,80);
s2 := Sym(81)!( 1,39)( 2,38)( 3,37)( 4,42)( 5,41)( 6,40)( 7,45)( 8,44)( 9,43)(10,12)(13,15)(16,18)(19,66)(20,65)(21,64)(22,69)(23,68)(24,67)(25,72)(26,71)(27,70)(28,29)(31,32)(34,35)(46,55)(47,57)(48,56)(49,58)(50,60)(51,59)(52,61)(53,63)(54,62)(73,74)(76,77)(79,80);
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, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
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*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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