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}*1344d
if this polytope has a name.
Group : SmallGroup(1344,11295)
Rank : 3
Schlafli Type : {8,4}
Number of vertices, edges, etc : 168, 336, 84
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,4}*672b
   4-fold quotients : {8,4}*336a
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 2.
      42 facets:
         42 of {8}*16
      84 vertex figures:
         84 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      44 facets:
         4 of {4}*8
         40 of {8}*16
      88 vertex figures:
         80 of {4}*8
         8 of {2}*4
   P/N, where N=<s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 3.
      28 facets:
         28 of {8}*16
      56 vertex figures:
         56 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2> of order 4.
      24 facets:
         6 of {4}*8
         18 of {8}*16
      48 vertex figures:
         36 of {4}*8
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s1> of order 4.
      24 facets:
         4 of {2}*4
         20 of {8}*16
      44 vertex figures:
         40 of {4}*8
         4 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
      16 facets:
         4 of {4}*8
         12 of {8}*16
      32 vertex figures:
         24 of {4}*8
         8 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 8.
      13 facets:
         5 of {4}*8
         8 of {8}*16
      26 vertex figures:
         16 of {4}*8
         10 of {2}*4
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2> of order 12.
      8 facets:
         6 of {8}*16
         2 of {4}*8
      16 vertex figures:
         4 of {2}*4
         12 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 24.
      5 facets:
         3 of {4}*8
         2 of {8}*16
      10 vertex figures:
         6 of {2}*4
         4 of {4}*8

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {8,4}

Twisty Puzzle