Character tables of p-groups with derived subgroup of prime order III

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. In this paper we compare the number of non-isomorphic p-groups with derived subgroup of order p with the number of non-isomorphic character tables of these groups. We show that the difference between the number of non-isomorphic groups of this type and the number of non-isomorphic character tables of these groups increases exponentially. Furthermore, we prove that if we fix the index of the center of these groups, say p2m, and we let the size of the groups grow bigger, then, for each character table there are on average (2m)!/(m2m!) non-isomorphic groups whose character table are isomorphic to the given one.