Polytope of Type {4,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,3490)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 108, 324, 162
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   27-fold quotients : {4,6}*48c
   54-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      81 facets:
         81 of {4}*8
      56 vertex figures:
         52 of {6}*12
         4 of {3}*6
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      81 facets:
         81 of {4}*8
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      54 facets:
         54 of {4}*8
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      54 facets:
         54 of {4}*8
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      54 facets:
         54 of {4}*8
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
      27 facets:
         27 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 6.
      27 facets:
         27 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 6.
      27 facets:
         27 of {4}*8
      20 vertex figures:
         16 of {6}*12
         4 of {3}*6
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 6.
      27 facets:
         27 of {4}*8
      20 vertex figures:
         16 of {6}*12
         4 of {3}*6
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
      27 facets:
         27 of {4}*8
      20 vertex figures:
         16 of {6}*12
         4 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 9.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 9.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 9.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 18.
      9 facets:
         9 of {4}*8
      6 vertex figures:
         6 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 18.
      9 facets:
         9 of {4}*8
      8 vertex figures:
         4 of {6}*12
         4 of {3}*6
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1, s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 18.
      9 facets:
         9 of {4}*8
      8 vertex figures:
         4 of {6}*12
         4 of {3}*6
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s0*s1*s0*s2*s1*s0*s2*s1*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 18.
      9 facets:
         9 of {4}*8
      8 vertex figures:
         4 of {6}*12
         4 of {3}*6

Permutation Representation (GAP) :

s0 := ( 1, 7)( 2, 9)( 3, 8)( 4,10)( 5,12)( 6,11);;
s1 := ( 7,10)( 8,11)( 9,12);;
s2 := ( 1, 2)( 4,11)( 5,12)( 6,10)( 7, 9);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(12)!( 1, 7)( 2, 9)( 3, 8)( 4,10)( 5,12)( 6,11);
s1 := Sym(12)!( 7,10)( 8,11)( 9,12);
s2 := Sym(12)!( 1, 2)( 4,11)( 5,12)( 6,10)( 7, 9);
poly := sub<Sym(12)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 >;

References : None.
to this polytope

Polytope of Type {4,6}

Twisty Puzzle