The group $PSL_2(\mathbb{F}_7)\cong GL_3(\mathbb{F}_2)$ has a pair of (complex) three dimensional representations, and I am wondering if anyone knows an explicit construction of these. To clarify what I mean by explicit, we can recognise the $6$ and $7$ irreducible dimensional representations being associated to the $2$-transitive actions on the fano plane and projective line over $\mathbb{F}_7$ respectively, and the $8$ dimensional representation is induced from a nontrivial linear rep of the index $8$ subgroup $C_7\rtimes C_3$. Its also the Steinberg representation, which I am not really familiar with, but this also counts as an "explicit description".
From this, one can fill out the character table and find the characters of the three dimensional reps, but given that their dimension is small, I would hope there's a nice construction of them.