Affine primitive permutation group ($G={\mathbb{F}}_p^d⋊H,H\le G{L}_d(p)\text{ irreducible}$)

An elementary abelian regular minimal normal subgroup acts as translations of a vector space. Its point stabilizer acts faithfully and irreducibly on that space; invariant subspaces would be blocks. The full affine normalizer is $AG{L}_d(p)$. In degree sixteen it lies inside $A_{16}$: nonzero translations swap eight pairs, and elementary linear transvections swap four pairs. Thus it is not a maximal subgroup of $S_{16}$.

The group of maps $x\mapsto Ax+b$ of a finite-dimensional vector space, with $A$ invertible. Identifying the underlying points with a regular elementary abelian group identifies its normalizer in the symmetric group with this semidirect product.