The main goal of this work is to examine classes of finite groups in which normality, permutability and Sylow-permutability are transitive relations. These classes of groups are called T , PT and PST , respectively. The main focus is on direct products of T , PT and PST groups and the behavior of a collection of cyclic normal, permutable and Sylow-permutable subgroups under the intersection map. In general, a direct product of finitely many groups from one of these classes does not belong to the same class, unless the orders of the direct factors are relatively prime. Examples suggest that for solvable groups it is not required to have relatively prime orders to stay in the class. In addition, the concept of normal, permutable and S-permutable cyclic sensitivity is tied with that of Tc, PTc and PSTc groups, in which cyclic subnormal subgroups are normal, permutable or Sylow-permutable. In the process another way of looking at the Dedekind, Iwasawa and nilpotent groups is provided as well as possible interplay between direct products and the intersection map is observed.