  We develop practical techniques to compute with arithmetic groups $H\leq
\mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a
principal congruence subgroup in $H$. Problems solved include testing
membership in $H$, analyzing the subnormal structure of $H$, and the
orbit-stabilizer problem for $H$. Effective computation with subgroups of
$\mathrm{GL}(n,\mathbb{Z}_m)$ is vital to this work. All algorithms have been
implemented in GAP.
