In a previous paper the authors described an algorithm to determine whether
a group of matrices over a finite field, generated by a given set of matri
ces, contains one of the classical groups or the special linear group. The
algorithm was designed to work for all sufficiently large field sizes and d
imensions of the matrix group. However, it did not apply to certain small c
ases. Here we present an algorithm to handle the remaining cases. The theor
etical background of the algorithm presented in this paper is a substantial
extension of that needed for the original algorithm.