  Superpermutations are words over a finite alphabet containing every
permutation as a factor. Finding the minimal length of a superpermutation is
still an open problem. In this article, we introduce superpermutations
matrices. We establish a link between the minimal size of such a matrix and the
minimal length of a universal word for the quotient of the symmetric group
$S_n$ by an equivalence relation. We will then give non-trivial bounds on the
minimal length of such a word and prove that the limit of their ratio when $n$
approaches infinity is 2.
