Finite groups generated in low real codimension

We study the intersection lattice of the arrangement AG of subspaces fixed by subgroups of a finite linear group G. When G is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of G. We generalize the notion of finite reflection groups. We say that a group G is generated (resp. strictly generated) in codimension k if it is generated by its elements that fix point-wise a subspace of codimension at most k (resp. precisely k).