Polytope of Type {4,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,9}*1296a
if this polytope has a name.
Group : SmallGroup(1296,3490)
Rank : 3
Schlafli Type : {4,9}
Number of vertices, edges, etc : 72, 324, 162
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {4,3}*48
   54-fold quotients : {4,3}*24
   108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      81 facets:
         81 of {4}*8
      36 vertex figures:
         36 of {9}*18
   P/N, where N=<s0*s1*s0*s1> of order 2.
      84 facets:
         6 of {2}*4
         78 of {4}*8
      36 vertex figures:
         36 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 3.
      54 facets:
         54 of {4}*8
      24 vertex figures:
         24 of {9}*18
   P/N, where N=<s1*s2*s1*s2*s1*s2> of order 3.
      54 facets:
         54 of {4}*8
      36 vertex figures:
         18 of {3}*6
         18 of {9}*18
   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
      24 vertex figures:
         24 of {9}*18
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
      45 facets:
         9 of {2}*4
         36 of {4}*8
      18 vertex figures:
         18 of {9}*18
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 6.
      30 facets:
         24 of {4}*8
         6 of {2}*4
      12 vertex figures:
         12 of {9}*18
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 6.
      30 facets:
         6 of {2}*4
         24 of {4}*8
      12 vertex figures:
         12 of {9}*18
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 6.
      27 facets:
         27 of {4}*8
      12 vertex figures:
         12 of {9}*18
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 6.
      27 facets:
         27 of {4}*8
      18 vertex figures:
         9 of {3}*6
         9 of {9}*18
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 6.
      27 facets:
         27 of {4}*8
      12 vertex figures:
         12 of {9}*18
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2> of order 9.
      18 facets:
         18 of {4}*8
      12 vertex figures:
         6 of {3}*6
         6 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 9.
      18 facets:
         18 of {4}*8
      16 vertex figures:
         4 of {9}*18
         12 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
      8 vertex figures:
         8 of {9}*18
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 18.
      12 facets:
         6 of {2}*4
         6 of {4}*8
      4 vertex figures:
         4 of {9}*18
   P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 18.
      9 facets:
         9 of {4}*8
      6 vertex figures:
         3 of {3}*6
         3 of {9}*18
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 18.
      9 facets:
         9 of {4}*8
      4 vertex figures:
         4 of {9}*18
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 18.
      9 facets:
         9 of {4}*8
      8 vertex figures:
         2 of {9}*18
         6 of {3}*6

Permutation Representation (GAP) :

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

References : None.
to this polytope

Polytope of Type {4,9}

Twisty Puzzle