I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in that case), the alternating group $\mathfrak{A}_n$ has a nontrivial central extension by $\mathbb{Z}/3$. The construction in the case $n=6$ is as follows.

Define an *oval* in $\mathbb{P}^2(\mathbb{F}_4)$ as a set of 6 points which are in general position (i.e. no 3 of them are collinear). It is easy to show that any projective frame of $\mathbb{P}^2(\mathbb{F}_4)$ is contained in a unique oval, and from this that the stabilizer of an oval in $PGL_3(\mathbb{F}_4)$ is isomorphic to $\mathfrak{A}_6$. The inverse image of this $\mathfrak{A}_6$ in $SL_3(\mathbb{F}_4)$ provides the nontrivial extension by $\mathbb{Z}/3$.

Does anyone know a reference for this? and/or an analogous construction for $\mathfrak{A}_7$?