  In this paper a method for embedding cocyclic submatrices with ``large''
determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is
described (t an odd integer). If these determinants attain the largest possible
value, we are embedding D-optimal designs. Applications to the pivot values
that appear when Gaussian Elimination with complete pivoting is performed on
these cocyclic Hadamard matrices are studied.
