  Let $n, k \geq 3$. In this paper, we analyse the quotient group
$B\_n/\Gamma\_k(P\_n)$ of the Artin braid group $B\_n$ by the subgroup
$\Gamma\_k(P\_n)$ belonging to the lower central series of the Artin pure braid
group $P\_n$. We prove that it is an almost-crystallographic group. We then
focus more specifically on the case $k=3$. If $n \geq 5$, and if $\tau \in N$
is such that $gcd(\tau, 6) = 1$, we show that $B\_n/\Gamma\_3 (P\_n)$ possesses
torsion $\tau$ if and only if $S\_n$ does, and we prove that there is a
one-to-one correspondence between the conjugacy classes of elements of order
$\tau$ in $B\_n/\Gamma\_3 (P\_n)$ with those of elements of order $\tau$ in the
symmetric group $S\_n$. We also exhibit a presentation for the
almost-crystallographic group $B\_n/\Gamma\_3 (P\_n)$. Finally, we obtain some
$4$-dimensional almost-Bieberbach subgroups of $B\_3/\Gamma\_3 (P\_3)$, we
explain how to obtain almost-Bieberbach subgroups of $B\_4/\Gamma\_3(P\_4)$ and
$B\_3/\Gamma\_4(P\_3)$, and we exhibit explicit elements of order $5$ in
$B\_5/\Gamma\_3 (P\_5)$.
