Given a smooth plane quartic curve C over a field $k$ of characteristic 0, with Jacobian variety $J$, and a marked rational point P $\in$ C($k$), we construct a reductive group $G$ and a $G$-variety $X$, together with an injection $J$($k$)/2$J$($k$) $\hookrightarrow$ $G$($k$)$\textit{X}$($\textit{k}$).WedothisusingtheMumfordthetagroupofthedivisor2$\Theta$ of $J$, and a construction of Lurie which passes from Heisenberg groups to Lie algebras.