We describe three different methods to compute all those characters of a finite group that have certain properties of transitive permutation characters. First, a combinatorial approach can be used to enumerate vectors of multiplicities. Secondly, these characters can be found as certain integral solutions of a system of inequalities. Thirdly, they are calculated via Gaussian elimination. The methods are used to determine these characters for some finite groups and runtimes are listed. In the final section, a permutation character of the Lyons group is constructed. (C) 1998 Academic Press.