Representing matrices of almost completely decomposable groups

Almost completely decomposable groups can be described in terms of integral matrices and in terms of anti-representations in finite modules over proper quotient rings of the ring of integers. The anti-representations are described by the so-called representating matrices. Representing matrices and their interrelationship with the integral matrices describing a group are studied in general. It is shown that the integral and representing matrices may be assumed to have a special form. Two applications demonstrate the usefulness of the results.