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|Tech Branch||Logistics & Ground Combat|
A Flag Bridge contains all the necessary facilities (large table for star charts, extra coffee machine, large jacuzzi, well-stocked bar) to enable a high-ranking fleet officer to properly command a Task Force.
In your home system, the Task Force leader is usually the officer in charge of Fleet Headquarters on your home planet, but the TF leader must be in the same system as the TGs under his command for his bonuses to apply. When a TF leaves your home system, having one ship with a flag bridge there can be very useful for training and combat initiative.
Note that you don't need a TF commander, it's just an added bonus. Also make sure the leader you want to be TF commander is assigned to the ship containing the bridge and that he's the highest ranking officer in the TF, because the highest ranking is in charge, even if he's not on the flagship, and if he isn't, he doesn't get the Flag Bridge bonus.