  There is always a natural embedding of $S_s\wr S_k$ into the linear
programming (LP) relaxation permutation symmetry group of an orthogonal array
integer linear programming (ILP) formulation with equality constraints.
  The point of this paper is to prove that in the $2$ level, strength $1$ case
the LP relaxation permutation symmetry group of this formulation is isomorphic
to $S_2\wr S_k$ for all $k$, and in the $2$ level, strength $2$ case it is
isomorphic to $S_2^k\rtimes S_{k+1}$ for $k\geq 4$. The strength $2$ result
reveals previously unknown permutation symmetries that can not be captured by
the natural embedding of $S_2\wr S_k$. We also conjecture a complete
characterization of the LP relaxation permutation symmetry group of the ILP
formulation.
