A tabadi was the senior female or matriarch of an Orion family. She was a counterpart to the tahedri, or patriarch.
The tabadi technically had only minimal say in family affairs, leaving these things to the tahedri. But in practice she could wield a great deal of influence, even if only by compromise and negotiation. On rare occasions, a retiring family leader could choose a female descendant to inherit, in which case she would have the full power of a tahedri. (FASA RPG modules: The Orions: Book of Common Knowledge, The Orions: Book of Deep Knowledge)