  Let $M$ be a compact surface without boundary, and $n\geq 2$. We analyse the
quotient group $B_n(M)/\Gamma_2(P_n(M))$ of the surface braid group $B_{n}(M)$
by the commutator subgroup $\Gamma_2(P_n(M))$ of the pure braid group
$P_{n}(M)$. If $M$ is different from the $2$-sphere $\mathbb{S}^2$, we prove
that $B_n(M)/\Gamma_2(P_n(M))$ is isomorphic rho $P_n(M)/\Gamma_2(P_n(M))
\rtimes_{\varphi} S_n$, and that $B_n(M)/\Gamma_2(P_n(M))$ is a
crystallographic group if and only if $M$ is orientable. If $M$ is orientable,
we prove a number of results regarding the structure of
$B_n(M)/\Gamma_2(P_n(M))$. We characterise the finite-order elements of this
group, and we determine the conjugacy classes of these elements. We also show
that there is a single conjugacy class of finite subgroups of
$B_n(M)/\Gamma_2(P_n(M))$ isomorphic either to $S_n$ or to certain Frobenius
groups. We prove that crystallographic groups whose image by the projection
$B_n(M)/\Gamma_2(P_n(M))\to S_n$ is a Frobenius group are not Bieberbach
groups. Finally, we construct a family of Bieberbach subgroups
$\tilde{G}_{n,g}$ of $B_n(M)/\Gamma_2(P_n(M))$ of dimension $2ng$ and whose
holonomy group is the finite cyclic group of order $n$, and if
$\mathcal{X}_{n,g}$ is a flat manifold whose fundamental group is
$\tilde{G}_{n,g}$, we prove that it is an orientable K\"ahler manifold that
admits Anosov diffeomorphisms.
