  Using a Mathematica code, we present a straightforward numerical analysis of
the 384-dimensional solution space of signed permutation 4x4 matrices, which in
sets of four provide representations of the GR(4,4) algebra, closely related to
the N=1 (simple) supersymmetry algebra in 4-dimensional spacetime. Following
after ideas discussed in previous papers about automorphisms and classification
of adinkras and corresponding supermultiplets, we make a new and alternative
proposal to use equivalence classes of the (unsigned) permutation group S4 to
define distinct representations of higher dimensional spin bundles within the
context of adinkras. For this purpose, the definition of a dual operator akin
to the well-known Hodge star is found to partition the space of these GR(4,4)
representations into three suggestive classes.
