Exceptional case (reviewed)

This is one of the six possible cases for the image of the mod $p$ Galois Representation if $p$ is a non-surjective prime for $E$.

The label S4 means that $G$ is the full pre-image in $\GL(2,\F_p)$ of a subgroup (unique up to conjugacy) of $\PGL(2,\F_p)$ that is isomorphic to the symmetric group on $4$ elements. This case can occur only if $p$ is congruent to $1$ modulo $4$.