Characterizing some groups with nilpotent derived subgroup

In this paper, groups with trivial intersection between Frattini and derived subgroups are considered. First, some structural properties of these groups are given in an important special case. Then, some family invariants of each n-isoclinism family of such groups are stated. In particular, an explicit bound for the order of each center factor group in terms of the order of its derived subgroup is also provided.