On the generic curve of genus 3

We study genus $g$ coverings of full moduli dimension of degree $d=[\frac {g+3} 2]$\mathrm{.}ThereisahomomorphismbetweenthecorrespondingHurwitzspace$\H$ofsuchcoverstothemodulispace$\M_g$ofgenus$g$curves\mathrm{.}Inthecase$g=3$, using the signature of such covering we provide an equation for the generic ternary quartic. Further, we discuss the degenerate subloci of the corresponding Hurwitz space of such covers from the computational group theory viewpoint. In the last section, we show that one of these degenerate loci corresponds to the locus of curves with automorphism group $C_3$. We give necessary conditions in terms of covariants of ternary quartics for a genus 3 curve to belong to this locus.