Quiver braid group action for a 3-fold crepant resolution

The 3-fold cyclic quotient singularity denoted $\frac{1}{7}(1,2,4)$ admits a crepant resolution X with three exceptional Hirzebruch surfaces intersecting pairwise along curves. We show that the derived category D(X) carries a faithful action of a quiver braid group, where the relevant quiver is a 3-cycle encoding the intersection data.