  For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of
genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible
plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a
field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are
known from the study of algebraic curves defined over a non-Archimedean
valuated field since for $|c|<1$ they are curves with a large solvable
automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from
the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal
with the case where $\mathbb{K}$ is an algebraically closed field of
characteristic $p$. We prove that the group $Aut(\mathcal{X})$ of all
automorphisms of $\mathcal{X}$ fixing $\mathbb{K}$ elementwise has order
$2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an
elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of
order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini
and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model
$\mathcal{Y}$ in the three-dimensional projective space $PG(3,\mathbb{K})$
which is neither classical nor Frobenius classical over the finite field
$\mathbb{F}_{q^2}$.
