  In this note, we give a so-called representative classification for the
strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$,
where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic
$p = 0$ or $p > 13$. We start with a classification already obtained by the
first author and we use standard techniques.
  Interestingly, in the way to get these families for the different strata, we
find two remarkable phenomenons that did not appear before. One is the
existence of a non $0$-dimensional final stratum of plane curves. At a first
sight it may sound odd, but we will see that this is a normal situation for
higher degrees and we will give a explanation for it.
  We explicitly describe representative families for all strata, except for the
stratum with automorphism group $\mathbb{Z}/5\mathbb{Z}$. Here we find the
second difference with the lower genus cases where the previous techniques do
not fully work. Fortunately, we are still able to prove the existence of such
family by applying a version of Luroth's theorem in dimension $2$.
