Polynilpotent multiplier of extra special P-groups

In this paper, we obtain some exact sequences on the polynilpotent multiplier of a group G. As a consequence of these exact sequences, we derive some numerical inequalities for their order, exponent, and generating set. One of the results is an explicit structure for the Baer invariant of extra special p-groups with respect to the variety of polynilpotent groups of class row (c1,...,ct), Nc1,...,ct. It has a useful application in order to prove that extra special p-groups have no any Nc1,c2-covering group, when c1 >= 2 and c2 > 2.