[member="Orphen"]
We don't have a core one: we have The Galactic Republic, which as usual have their Jedi, the Silver Jedi, which are a Major Faction in their own right, and then a cacophony of minor Jedi factions.
All of them are as valid as each other. If I were you I'd do a little research into each and see which one best suits your writing style and the character you have in mind. There will be subtle differences.
I recommend speaking to Corvus Raaf when it comes to the Republic, if you want anything clearing up, and Thurion Heavenshield or Iella E'ron for the Silver Jedi. As for the smaller Factions, seek out the Faction Owner (it shows beneath the faction name in the Faction Directory!)
EDIT: We also have the Levantine Sanctum, which was built from Former Jedi who wanted to make a difference without all the political poodoo that held back the Republic Jedi - amongst other things. They're pretty sweet.
As I say, depending on the character you have in mind, will alter which Faction he/she will be more suited to
