Polynilpotent multiplier of extra special p-groups

In this paper, we obtain some exact sequences on the polynilpotent multiplier of a group [Formula: see text]. 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 [Formula: see text]-groups with respect to the variety of polynilpotent groups of class row [Formula: see text], [Formula: see text]. It has a useful application in order to prove that extra special [Formula: see text]-groups have no any [Formula: see text]-covering group, when [Formula: see text] and [Formula: see text].