We develop a method for grouping the $2^k - 1$ factorial effects in a 2-level factorial design into mutually exclusive sets of the form $(s, t, st)$, where $st$ is the generalized interaction of effects $s$ and $t$. As an application, we construct orthogonal arrays $OA(2^k, 2^m4^n, 2)$ of size $2^k, m$ constraints with 2 levels and $n$ constraints with 4 levels in the construction cannot be further improved. In this sense our grouping scheme is optimal. We discuss the advantages of the present approach over other construction methods.