In dimension 2 these groups completely describe the hyperoctahedral group, which is the dihedral group Dih4 of order 8, and is an extension 2.V (of the 4-group by a cyclic group of order 2).
Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension:[REF] These groups have n orthogonal mirrors in n-dimensions.