Product action of a wreath product ($H\wr K\curvearrowright {\Delta }^r$)

The base group acts independently on each coordinate of a Cartesian power, and the top group permutes coordinates. This is distinct from the imprimitive action on a disjoint union of blocks. It is primitive when the base action is primitive and nonregular and the top group is transitive. The full product-action overgroup is $S_{|\Delta |}\wr S_r$. Socles of almost-simple, diagonal or holomorph bases produce respectively product-action, compound-diagonal or holomorph-compound types.