  The orbits in $\Gamma_{\infty}(3) \backslash \Gamma(3)$ are in bijection with
sets of invariants satisfying certain relations. We explain how wedge product
matrices give an alternative definition of the invariants of matrix orbits.
This new method provides the possibility of performing similar computations
with other congruence subgroups and arbitrary $n \times n$ matrices. Using
Steinberg's refined version of the Bruhat decomposition, we construct an
explicit choice of coset representative for each orbit in the orbit space
$\Gamma_{\infty}(3) \backslash \Gamma(3)$ of $3 \times 3$ matrices over the PID
of Eisenstein integers.
