On some Galois covers of the Suzuki and Ree curves

We investigate two families S˜q and R˜q of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. We show that S˜q is not Galois covered by the Hermitian curve maximal over Fq4, and R˜q is not Galois covered by the Hermitian curve maximal over Fq6. We also compute the genera of many Galois subcovers of S˜q and R˜q; in this way, many new values in the spectrum of genera of maximal curves are obtained. The full automorphism group of both S˜q and R˜q is determined.