Group rings of finite strongly monomial groups: Central units and primitive idempotents

We compute the rank of the group of central units in the integral group ring ZGZG of a finite strongly monomial group G. The formula obtained is in terms of the strong Shoda pairs of G  . Next we construct a virtual basis of the group of central units of ZGZG for a class of groups G properly contained in the finite strongly monomial groups. Furthermore, for another class of groups G   inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of QGQG.Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of ZGZG, this for metacyclic groups G   of the form G=Cqm⋊CpnG=Cqm⋊Cpn with p and q   different primes and the cyclic group CpnCpn of order pnpn acting faithfully on the cyclic group CqmCqm of order qmqm.