On some Galois covers of the Suzuki and Ree curves

We determine the full automorphism group of two recently constructed families ${\tilde{\mathcal{S}}}_q$ and ${\tilde{\mathcal{R}}}_q$ of maximal curves over finite fields. These curves are covers of the Suzuki and Ree curves, and are analogous to the Giulietti-Korchm'aros cover of the Hermitian curve. We also show that ${\tilde{\mathcal{S}}}_q$ is not Galois covered by the Hermitian curve maximal over ${\mathbb{F}}_{q^4}$, and ${\tilde{\mathcal{R}}}_q$ is not Galois covered by the Hermitian curve maximal over ${\mathbb{F}}_{q^6}$. Finally, we compute the genera of many Galois subcovers of ${\tilde{\mathcal{S}}}_q$ and ${\tilde{\mathcal{R}}}_q$; this provides new genera for maximal curves.