A parallel flats fraction for the $3^n$ design is defined as union of flats ${t}At=c_i(mod 3)}, i=1,2,cdots, f$ and is symbolically written as At=C where A is rank r. The A matrix partitions the effects into n+1 alias sets where $u=(3^{n-r}-1)/2. For each alias set the f flats produce an ACPM from which a detection matrix is constructed. The set of all possible parallel flats fraction C can be partitioned into equivalence classes. In this paper, we develop some properties of a detection matrix and C.