Polytope of Type {8,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4}*672e
if this polytope has a name.
Group : SmallGroup(672,1254)
Rank : 3
Schlafli Type : {8,4}
Number of vertices, edges, etc : 84, 168, 42
Order of s₀s₁s₂ : 14
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {8,4,2} of size 1344
Vertex Figure Of :
   {2,8,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,4}*336b
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,4}*1344f
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      22 facets:
         20 of {8}*16
         2 of {4}*8
      44 vertex figures:
         40 of {4}*8
         4 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
      12 facets:
         9 of {8}*16
         3 of {4}*8
      24 vertex figures:
         18 of {4}*8
         6 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
      8 facets:
         6 of {8}*16
         2 of {4}*8
      16 vertex figures:
         12 of {4}*8
         4 of {2}*4

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {8,4}

Twisty Puzzle