Polytope of Type {9,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*1296e
if this polytope has a name.
Group : SmallGroup(1296,3490)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 108, 324, 72
Order of s₀s₁s₂ : 4
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) :
   27-fold quotients : {3,6}*48
   54-fold quotients : {3,3}*24
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*s1*s0*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1> of order 3.
      36 facets:
         18 of {3}*6
         18 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 9.
      12 facets:
         6 of {3}*6
         6 of {9}*18
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 9.
      16 facets:
         4 of {9}*18
         12 of {3}*6
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 9.
      8 facets:
         8 of {9}*18
      12 vertex figures:
         12 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {9,6}

Twisty Puzzle