  Let F/k be a Galois extension of number fields with dihedral Galois group of
order 2q, where q is an odd integer. We express a certain quotient of S-class
numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit
index. Our formula is valid for arbitrary extensions with Galois group D_{2q}
and for arbitrary Galois-stable sets of primes S, containing the Archimedean
ones. Our results have curious applications to determining the Galois module
structure of the units modulo the roots of unity of a D_{2q}-extension from
class numbers and S-class numbers. The techniques we use are mainly
representation theoretic and we consider the representation theoretic results
we obtain to be of independent interest.
