  We give new contributions to the existence problem of canonical surfaces of
high degree. We construct several families (indeed, connected components of the
moduli space) of surfaces $S$ of general type with $p_g=5,6$ whose canonical
map has image $\Sigma$ of very high degree, $d=48$ for $p_g=5$, $d=56$ for
$p_g=6$. And a connected component of the moduli space consisting of surfaces
$S$ with $K^2_S = 40, p_g=4, q=0$ whose canonical map has always degree $\geq
2$, and, for the general surface, of degree $2$ onto a canonical surface $Y$
with $K^2_Y = 12, p_g=4, q=0$.
  The surfaces we consider are SIP 's, i.e. surfaces $S$ isogenous to a product
of curves $(C_1 \times C_2 )/ G$; in our examples the group $G$ is elementary
abelian, $G = (\mathbb{Z}/m)^k$. We also establish some basic results
concerning the canonical maps of any surface isogenous to a product, basing on
elementary representation theory.
