  In 1895 Wiman introduced a Riemann surface $\mathcal{W}$ of genus $6$ over
the complex field $\mathbb{C}$ defined by the homogeneous equation
$\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2=0$, and
showed that its full automorphism group is isomorphic to the symmetric group
$S_5$. The curve $\mathcal{W}$ was previously studied as a curve defined over a
finite field $\mathbb{F}_{p^2}$ where $p$ is a prime, and necessary and
sufficient conditions for its maximality over $\mathbb{F}_{p^2}$ were obtained.
In this paper we first show that the result of Wiman concerning the
automorphism group of $\mathcal{W}$ holds also over an algebraically closed
field $\mathbb{K}$ of positive characteristic $p$, provided that $p \geq 7$.
For $p=2,3$ the polynomial $X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2
Z^2$ is not irreducible over $\mathbb{K}$, while for $p=5$ the curve
$\mathcal{W}$ is rational and $Aut(\mathcal{W}) \cong PGL(2,\mathbb{K})$. We
also show that the $\mathbb{F}_{19^2}$-maximal Wiman's sextic $\mathcal{W}$ is
not Galois covered by the Hermitian curve $\mathcal{H}_{19}$ over
$\mathbb{F}_{19^2}$.
