A parallel flats faraction for the $3^n$ factorial experiment is symbolically written as $At = C(r imesf)$ where $A(r imesn)$ is of rank r. The A-matrix partitions the nonnegligible effects into $(3^{n-r}-1)/2+1$ alias sets. The $U_i$ effects in the i-th alias set are related pairwise by elements from $S_3$, the symmetric group on three symbols. For each alias set the f flats produce an $f imes u_i$ alias componet permutation matrices (ACPM) with elements from $S_3$. All the information concerning the relationships among levels of the effects is contained in the ACPM.