Non-very generic arrangements in low dimension

Discriminantal arrangements introduced by Manin and Schectmanin in $1989$ are induced arrangements in the space of all translations of an original arrangement $mathcal{A}$. The combinatorics of the discriminantal arrangement depends on the original arrangement $mathcal{A}$ therefore in 1997 Bayer and Brandt defined two classes of original arrangements. One consists of the arrangements such that the intersection lattice of the induced discriminantal arrangement has maximum cardinality. Those arrangements are called extit{very generic}. The others are called extit{non-very generic}. In this paper, we will provide both a characterization and a classification of few non-very generic arrangements in low dimensional space in terms of permutation groups.