3 factions? The Fallen has 5 houses; Devils, Exile, Wolves, Winter and Kings
The Cabal has 7 [i]known[/i] detachments: Blind Legion, The Skyburners, Siege Dancers, Ice Reapers, Bone Crushers, Dust Giants and Sand Eaters
The Vex has 8 programmings: Aphix Invasive, Sol Imminent, Hezen Corrective, Hezen Protective, Sol Divisive, Sol Primeval, Sol Progeny and the Virgo Prohibition.
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I just found that out looking at the Grimoire cards. Sorry for my inaccuracy.
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You have nothing to be sorry for...I'm merely pointing it out for the benefit of education :)