You searched for: “metrocracy
metrocracy (s) (noun), metrocracies (pl)
Any community, society, or social group that is based on matriarchy: Metrocracy is a form of organization or government where women have power.

Metrocracy can be further explained as a form of social order where women are in charge and are recognized as the heads of families, with power, lineage, and inheritance passing, where possible, from mothers to daughters.