  Small Coxeter groups are precisely the ones for which the Tits representation
is integral, which makes the study of their congruence subgroups relevant. The
symmetric group $S_n$ has three natural extensions, namely, the braid group
$B_n$, the twin group $T_n$ and the triplet group $L_n$. The latter two groups
are small Coxeter groups, and play the role of braid groups under the
Alexander-Markov correspondence for appropriate knot theories, with their pure
subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane
spaces. In this paper, we prove that the congruence subgroup property fails for
infinite small Coxeter groups which are not virtually abelian. As an
application, we deduce that the congruence subgroup property fails for both
$T_n$ and $L_n$ when $n \ge 4$. We also determine subquotients of principal
congruence subgroups of $T_n$, and identify the pure twin group $PT_n$ and the
pure triplet group $PL_n$ with suitable principal congruence subgroups.
Further, we investigate crystallographic quotients of these two families of
small Coxeter groups, and prove that $T_n /PT_n^{'}$, $T_n/T_n^{''}$ and $L_n
/PL_n^{'}$ are crystallographic groups. We also determine crystallographic
dimensions of these groups and identify the holonomy representation of
$T_n/T_n^{''}$.
