[Formerly: Publ. I.R.M.A. Strasbourg, 1992, 473/S-27, p. 81-100.]

Based on this approach we present an algorithm for the construction of
a system of representatives of designs with given
parameters *t-(v,k,\lambda)* and a given automorphism
group *A*.

Firstly we present a method for computing the incidence matrices
*M ^{A}_{t,k}* by means of double cosets.

Solving the above system of equations is an NP-complete problem. We use a heuristic approach and represent the set of all solutions implicitly by a graph. This gives us the possibility either to extract the solutions explicitly, if there are not too many of them, or to compute their numbers.

Finally we can construct the isomorphism types of *t*-designs
with given parameters and given automorphism
group *A*, if we know about the structure
of overgroups of *A*, or, if there are too
many designs, we are in many cases still able to
give the precise numbers.

With the help of the complete algorithm we verify many prominent results. To the best of our knowledge our approach for the first time allowed to compute the precise number of isomorphism types or even these designs themselves for substantial numbers; see the examples and tables at the end of this publication.

Lehrstuhl II für Mathematik, Universität Bayreuth

An extended version of this paper has been
finally published under the title
"The *t*-designs with prescribed automorphism group,
new simple 6-designs" in
*J. Combin. Des.* **1** (1993), 125-170.