On Riemann Surfaces of genus g with 4g automorphisms

We determine, for all genus $g\ge 2$ the Riemann surfaces of genus $g$ with $4g$ automorphisms. For $g\mathrm{\ne }$ $3,6,12,15$ 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.