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A (projective, geometrically irreducible, non-singular) curve $\mathcal{X}$defined over a finite field $\mathbb{F}_{q^2}$ is maximal if the number$N_{q^2}$ of its $\mathbb{F}_{q^2}$-rational points attains the Hasse-Weilupper bound, that is $N_{q^2}=q^2+2\mathfrak{g}q+1$ where $\mathfrak{g}$ is thegenus of $\mathcal{X}$. An important question, also motivated by applicationsto algebraic-geometry codes, is to find explicit equations for maximal curves.For a few curves which are Galois covered of the Hermitian curve, this has beendone so far ad hoc, in particular in the cases where the Galois group has primeorder. In this paper we obtain explicit equations of all Galois covers of theHermitian curve with Galois group of order $p^2$ where $p$ is thecharacteristic of $\mathbb{F}_{q^2}$. Doing so we also determine the$\mathbb{F}_{q^2}$-isomorphism classes of such curves and describe their full$\mathbb{F}_{q^2}$-automorphism groups.

Galois subcovers of the Hermitian curve in characteristic $p$ with respect to subgroups of order $p^2$