  We prove that the minimal representation dimension of a direct product $G$ of
non-abelian groups $G_1,\ldots,G_n$ is bounded below by $n+1$ and thereby
answer a question of Ab\'ert. If each $G_i$ is moreover non-solvable, then this
lower bound can be improved to be $2n$. By combining this with results of
Pyber, Segal, and Shusterman on the structure of boundedly generated groups we
show that branch groups cannot be boundedly generated.
