  In the present paper we study structural aspects of certain quotients of
braid groups and virtual braid groups. In particular, we construct and study
linear representations $B_n\to {\rm
GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$, $VB_n\to {\rm
GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}, t_1^{\pm1},t_2^{\pm1},\ldots,
t_{n-1}^{\pm1}]\right)$ which are connected with the famous
Lawrence-Bigelow-Krammer representation. It turns out that these
representations are faithful representations of crystallographic groups
$B_n/P_n'$, $VB_n/VP_n'$, respectively. Using these representations we study
certain properties of the groups $B_n/P_n'$, $VB_n/VP_n'$. Moreover, we
construct new representations and decompositions of universal braid groups
$UB_n$.
