We characterize a natural class of modular categories of prime power
Frobenius-Perron dimension as representation categories of twisted doubles of
finite p-groups. We also show that a nilpotent braided fusion category C admits
an analogue of the Sylow decomposition. If the simple objects of C have
integral Frobenius-Perron dimensions then C is group-theoretical. As a
consequence, we obtain that semisimple quasi-Hopf algebras of prime power
dimension are group-theoretical. Our arguments are based on a reconstruction of
twisted group doubles from Lagrangian subcategories of modular categories (this
is reminiscent to the characterization of doubles of quasi-Lie bialgebras in
terms of Manin pairs).
