In most cases the name of the faction includes the name of its home system. Each faction has a home system from which they cannot retreat. They control stations and markets, and hand out missions. Minor factions are political entities within Elite’s universe. One of them, usually the biggest, determines control of the whole system, which means that if a faction controls that asset, they’ll control the system as well. There are several kinds: Coriolis, Orbis, planetary, etc. You can increase your reputation by doing missions for that faction, turning in bounties issued by them, trading in the markets they control (including black markets) and selling exploration data in stations they control.
The higher your reputation, the better the missions they’ll give you. With these ticks, influence values and faction states are updated according to actions performed by players.
