Mod-$\ell$ Galois image of a genus 2 curve (awaiting review)

Let $\ell$ be a prime and let $C$ be an genus 2 curve over a number field $K$ with Jacobian $A$.

Subgroups $G$ of $\GSp(4,\F_\ell)$ that can arise as the image of the mod-$\ell$ Galois representation \[ \bar\rho_{A,\ell}\colon {\Gal}(\overline{K}/K)\to \GSp(4,\F_\ell) \] attached to $A$ that do not contain $\Sp(4,\F_\ell)$ are identified up to conjugacy using labels of the form \[ \texttt{L.i.n}, \] where $\texttt{L}$ is the prime $\ell$, $\texttt{i}$ is the index of $G$ in $\GSp(4,\F_\ell)$, and $\texttt{n}$ is a positive integer that distinguishes subgroups of $\GSp(4,\F_\ell)$ of the same index. Clicking on such a label will provide more details about $G$, including generators that can be used to construct it.