  We define and study two structures associated to permutation groups:
Dirichlet characters on permutation groups, and the "cycle form," a bilinear
form on the group algebras of permutation groups. We use Dirichlet characters
and the cycle form to find a new upper bound on the number of unlabelled
bicolored graphs with $p$ red vertices and $q$ blue vertices. We use this bound
to calculate the asymptotic growth rate of the number of such graphs as
$p,q\rightarrow\infty$, answering a 1973 question of Harrison in the case where
$q-p$ is fixed. As an application, we show that, in an asymptotic sense, "most"
elements of the power set $P(\{ 1, \dots ,p\} \times \{ 1, \dots ,q\})$ are in
free $\Sigma_p\times \Sigma_q$-orbits.
