I am interested in understanding the representation theory of certain finite nilpotent groups (via the complex numbers). The groups $ G $ of interest have the following properties:

1) G is $ 3 $step nilpotent finite group

2) G is a $ 2 $-Group

3) G is an extension of a 2-stage nilpotent group H by $ mathbb {Z} / 2 mathbb {Z} $,

I know that the irreducible representations of a finite nilpotent group are monomial, and there is a unified construction of these representations in the case of $ 2 $step nilpotent groups. For a $ 2 $step nilpotent group $ H $, any irreducible representation $ rho $ of $ H $ the dimension greater than $ 1 $ is induced from a character of a fixed maximum Abelian subgroup $ S $ of $ H $, Actually $ rho $ is determined by its central character.

Question:

What can be said about the representation theory of $ G $? Can we identify a minimal set of subgroups that can be used to construct each irreducible representation as a monomial representation? (For example for a $ 2 $step nilpotent group We need a maximum Abelian subgroup and the group itself.)

I would appreciate thoughts on this question and references to 1-3 above. That is, all references to the representation theory of $ 3 $-step nilpotent groups, $ 2 $-Groups or double extensions of $ 2 $step nilpotent group or any combination thereof.

Sincerely yours,

Moshe