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

Algorithms for arithmetic groups with the congruence subgroup property