I would like to know if there is a systematic way to decompose direct products of irreps of a wreath product of the orthogonal group. I'm talking about the group $G_{m,n}=O(m)\wr S_n=O(m)^n\rtimes S_n$. For example, I know that $G_{m,n}$ has an irrep of dimension $mn$ furnished by a vector $\phi_i$, and an irrep of dimension $n-1$ furnished by a traceless-symmetric matrix $X_{ij}$, where $i,j=1,\ldots,mn$. Is there a systematic way to construct the decomposition of the direct product of these two irreps, i.e. to compute $\phi_i\otimes\phi_j$, $\phi_i\otimes X_{jk}$ and $X_{ij}\otimes X_{kl}$?
For a wreath product of the form $G_n=\Gamma\wr S_n=\Gamma^n\rtimes S_n$ with $\Gamma$ a finite group the answer to the question above appears to be affirmative, but it relies on the fact that $\Gamma$, being a finite group, has a finite number of irreps.