Affine, almost simple, diagonal, product action, and twisted wreath types

For a finite primitive permutation group $G\le \mathrm{Sym}(\mathrm{\Omega })$, the five coarse O'Nan-Scott types used on this page are:

• Affine type: the socle is the unique minimal normal subgroup, it is abelian and regular, and A unique abelian minimal normal subgroup gives affine type identifies it with a finite vector space.
• Almost simple type: the socle is a nonabelian simple group and the whole group lies between that socle and its full automorphism group in the sense of Almost simple finite groups.
• Diagonal type: the socle is a direct product $T^k$, with $k\ge 2$, of isomorphic nonabelian simple groups, and the action is the standard diagonal action on a coset space of a diagonal subgroup.
• Product action type: after identifying $\mathrm{\Omega }$ with ${\mathrm{\Delta }}^{\mathrm{\ell }}$ for some $\mathrm{\ell }\ge 2$, there is a primitive group $H$ on $\mathrm{\Delta }$ of almost simple or diagonal type, with $N=\mathrm{Soc}(H)$, such that $N^{\mathrm{\ell }}=\mathrm{Soc}(G)\le G\le H\wr K,$ where $K\le S_{\mathrm{\ell }}$ is the transitive group induced by $G$ on the coordinates and the wreath product has its product action. If $(h_1,\dots ,h_{\mathrm{\ell }};k)\in H^{\mathrm{\ell }}⋊K$, its product action is $({\delta }_1,\dots ,{\delta }_{\mathrm{\ell }})⟼({\delta }_{k^{-1}(1)}{h}_{k^{-1}(1)},\dots ,{\delta }_{k^{-1}(\mathrm{\ell })}{h}_{k^{-1}(\mathrm{\ell })})\mathrm{.}$
• Twisted wreath type: the socle is again regular and nonabelian, but the regular action is built from a twisted wreath product rather than from an abelian vector-space action.