Here is the full list of all 14 HEXTERMINATE achievements.
Remade the Empire, conquering all factions.
Defeated Navarre Hexer, the Pirate leader.
Blocked a Particle Accelerator with the Phase Barrier.
Installed a shield module on your ship.
Eliminated the Ascent from the galaxy.
Left a sector with less than 25% of your bridge's health.
Found your own Particle Accelerator.
Requested an Arbellatris battleship to join your fleet.
Escaped the Cradle of Isaldren.
Assembled a dreadnought.
Defeated the Chrysamere.
Destroyed the Chrysamere for good.
Defeated the Marauders.
Conquered Iriani Prime without the help of other Imperial ships.