On Riemann Surfaces of genus g with 4g automorphisms

We determine, for all genus $g\geq2$ the Riemann surfaces of genus $g$ with $4g$ automorphisms. For $g\neq$ $3,6,12,15$ or $30$, this surfaces form a real Riemann surface $\mathcal{F}_{g}$ in the moduli space $\mathcal{M}_{g}$: the Riemann sphere with three punctures. The set of real Riemann surfaces in $\mathcal{F}_{g}$ consists of three intervals its closure in the Deligne-Mumford compactification of $\mathcal{M}_{g}$ is a closed Jordan curve.