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Galen never fit in on Gallifrey. He likes excitement, he likes challenges, he likes honesty, he likes honorable plain dealing, and he likes loyalty. None of which, as he's happy to tell anyone who'll listen, are easy to find on the planet of the Time Lords. He was almost expelled from the Academy more than once, despite his undoubted technical aptitude. Truancy was the biggest issue - on several occasions, he sneaked out of the Capitol for extended periods to socialize with the Outsiders, who taught him a great deal about physical combat and desert survival. He was tempted to join them for good, but he loved technology too much to be content with a more primitive existence.
He graduated three days before the Sontarans invaded Gallifrey, and he was still living in his quarters at the Academy when it was besieged by Sontaran shock troops. Galen led a group of his fellow students in a breakout, using improvised weapons made from the materials that he had at hand in his workroom. He knew that Castellan Kelner would have every TARDIS in the Capitol locked down, so he headed for the decommissioning bay, gambling that there might be at least one TARDIS there that was still operational.
He won the gamble. In what he considered a fitting piece of poetic justice, he used the old Type 49 he found to steal a supply of weapons from the Sontaran command cruiser orbiting Gallifrey. Unwilling to risk sneaking aboard, he used a spacial overlap maneuver to carve a chunk of the armory directly out of the vessel's superstructure, inflicting substantial damage on the ship in the process. Then he delivered the Sontaran guns to an Outsider camp near the Capitol. While he led an Outsider raiding party - now heavily armed - back into the Capitol to deal with the Sontaran advance force, he programmed the stolen TARDIS to travel to Rutan space and transmit a warning about the Sontaran invasion. The Rutans couldn't risk the Sontarans getting their hands on Time Lord technology, he reasoned; the moment they knew what was happening, they'd attack the fleet orbiting Gallifrey with everything they had, leaving the Sontaran landing force cut off from any hope of reinforcements.
By the time Galen and his allies reached the Capitol, the Doctor had already used the D-Mat gun to kill the Sontaran commander and defeat the invasion, but Galen's plan might have worked – a fact that wasn't lost on some of the more cynical and self-serving members of the High Council. While most of the planet's leadership had spent the invasion either hiding or collaborating, Galen had struck back. Worse, he'd struck back effectively. The question that a lot of powerful Time Lords didn't want asked was very simple – if a newly graduated, borderline-delinquent student could do that, why couldn't you, my Lord Cardinal?
So they exiled him. Although the exile they intended - a thousand years of bucolic stagnation in an uninhabited galactic backwater - wasn't quite the one he ended up with. With the aid of his new K9 unit - secretly constructed with the clandestine help of the outgoing Lord President - Galen hijacked the TARDIS that was carrying him into exile, crash-landing it on the Sontaran-besieged Rutan colony where he'd sent the stolen Type 49. With a little careful timing, and the unwitting assistance of an orbiting Sontaran battle station, he managed to fake his own death. The High Council - now doubly embarrassed by his actions - weren't especially interested in questioning his supposed demise.
At his core, Galen is all about fixing things. On Gallifrey, the "things" were mostly technological, because there was nothing about Gallifrey that its inhabitants were willing to admit needed fixing - at least until the Sontarans invaded. He's discovered that the wider universe is filled with far more difficult challenges than any he faced on his home world, but he's still learning that injustice and oppression are a lot harder to fix than a broken circuit. He's an uncomplicated soul - he wants to enjoy his freedom and do some good in the process, without any deeper agenda.
He's friendly, he doesn't give himself airs, and his genuine willingness to help out wins him friends, but he's also slightly taciturn, direct to the point of blunt tactlessness, and has the kind of honesty that'll get you a straight answer if you ask him if your butt looks big in this. Essentially, he's a very clever but socially clueless nerd, with a heart of gold, but also a nerd's stereotypical quality of being more comfortable with computers than people.