  In this article we explicitly determine the Weierstrass semigroup at any
point and the full automorphism group of a known $\mathbb{F}_{q^2}$-maximal
curve $\mathcal{X}_3$ having the third largest genus. This curve arises as a
Galois subcover of the Hermitian curve, and its uniqueness (with respect to the
value of its genus) is a well-known open problem. Knowing the Weierstrass
semigroups may provide a key towards solving this problem. Surprisingly enough
$\mathcal{X}_3$ has many different types of Weierstrass semigroups and the set
of its Weierstrass points is much richer than the set of
$\mathbb{F}_{q^2}$-rational points, as instead happens for all the known
maximal curves where the Weierstrass points are known. We show that a similar
exceptional behaviour does not occur in terms of automorphisms, that is,
$\mathrm{Aut}(\mathcal{X}_3)$ is exactly the automorphism group inherited from
the Hermitian curve, apart from small values of $q$.
