  Let {\Lnk} be the class of all $n$-dimensional real solvable Lie algebras
having $k$-dimensional derived ideals. In 2020 the authors et al. gave a
classification of all non 2-step nilpotent Lie algebras of {\Li}. We propose in
this paper to study representations of these Lie algebras as well as their
corresponding connected and simply connected Lie groups. That is, for each
algebra, we give an upper bound of the minimal degree of a faithful
representation. Then, we give a geometrical description of coadjoint orbits of
corresponding groups. Moreover, we show that the characteristic property of the
family of maximal dimensional coadjoint orbits of a MD-group studied by K. P.
Shum and the second author et al. is still true for the Lie groups considered
here. Namely, we prove that, for each considered group, the family of the
maximal dimensional coadjoint orbits forms a measurable foliation in the sense
of Connes. The topological classification of these foliations is also provided.
