The Division

The Division (or simply Division) was a shadowy rogue Time Lord organisation from Gallifrey who believed they could "guide and shape events" across space and time as they saw fit, directly contraventing the official policy of the Time Lords "strict non-interference" and evaded them by existing "outside one universe and on the cusp of many more" (TV: Survivors of the Flux)

By the Thirteenth Doctor's time, with Gallifrey gone, (TV: The Timeless Children) the only known surviving operative was Karvanista, (TV: The Halloween Apocalypse) who had fought alongside the Fugitive Doctor. (TV: Once, Upon Time) However, the Division was still very much active, (TV: Village of the Angels) and had begun a plan to destroy the universe in response to the Doctor learning about them. (TV: Survivors of the Flux)

Early history
Officially, the Division did not exist nor had operatives, and its history was mostly redacted from the Matrix. (TV: The Timeless Children) The Division began on Gallifrey, but outgrew the Time Lords after they swore a policy of non-intervention and sought to control the universe. (TV: Survivors of the Flux)

An early incarnation of the Doctor, during their period as the Timeless Child, was contacted, tested and recruited into the Division by Solpado. (TV: The Timeless Children) During her service to the Division, the Fugitive Doctor was sent alongside Karvanista and two other Division agents to capture Swarm and Azure at the Temple of Atropos. This was the Siege of Atropos, in which the Time Lords' structure for Time was restored.

In the aftermath, Swarm was successfully imprisoned. (TV: Once, Upon Time) He was kept on the Burnished Rage battleground, with Division agents visiting once every thousand years to check he was secure. En Sentac was an agent who served on many of these tours. (TV: The Halloween Apocalypse)

The Fugitive Doctor at some point tried to escape from service to the Division, hiding out on Earth under a Chameleon Arch to avoid detection. Division agent Gat employed a troop of Judoon to capture her. They traced the Doctor to Gloucester in 2020, but the Doctor tricked Gat into shooting herself, and escaped in her TARDIS. (TV: Fugitive of the Judoon)

Eventually, the Division thanked the Doctor for their service, then wiped their memory. (TV: The Timeless Children)

Thirteenth Doctor's investigations
In the fictional story of Brendan in the Matrix, one of the clues to the truth was that the carriage clock he was given upon his retirement was dedicated for "services to the Division" instead of the Garda.

Eventually, a Judoon cold case unit transmatted inside the Thirteenth Doctor's TARDIS and imprisoned her inside a maximum security facility for her crimes against the Division, (TV: The Timeless Children) as well as 7000 other crimes, though she ultimately escaped. (TV: Revolution of the Daleks)

After learning the truth of her forgotten lives in service to the Division, (TV: The Timeless Children) the Thirteenth Doctor sought out Karvanista for information on the organisation, as he was the only surviving Division agent she could find. (TV: The Halloween Apocalypse) At this time the Division, by now led by Tecteun, decided to abandon the universe for the next one, instigating a plan to destroy the universe, (TV: Survivors of the Flux) by releasing the Flux in space and the Ravagers, Swarm and Azure, in time. (TV: Once, Upon Time)

En Sentac served her final tour, checking Swarm was secure in his containment chamber. Swarm found his opportunity to breach the chamber, and killed her, allowing him to regenerate into a new form. Swarm then freed his sister, (TV: The Halloween Apocalypse) and they proceeded to overtake the Temple of Atropos once more, (TV: War of the Sontarans) triggering the Great Disruption. At the heart of the time storm, the Thirteenth Doctor relived the original siege within her time stream, providing some answers to her questions. After she restored the Mouri and repaired time, Tecteun contacted the Doctor to warn her off further attempts to stop the inevitable end of the universe. (TV: Once, Upon Time)

After one Weeping Angel went rogue with knowledge of the Division's entire history, an Extraction Squad of Angels was sent to retrieve it. The rogue Angel made a deal to hand over the Thirteenth Doctor to the squad instead, with her being recalled to the Division. (TV: Village of the Angels)

Behind the scenes

 * Though no link has been drawn in-story, the creation of a secret Gallifreyan organisation with authority to break the Time Lords' non-interference protocols, overseen by one of Gallifrey's Founders, is strikingly similar to the depiction of the origins of the Celestial Intervention Agency under the helm of Rassilon in such stories as The Legacy of Gallifrey. In turn, the Fugitive Doctor acting as a reluctant Division agent in a forgotten era of the Doctor's life in Fugitive of the Judoon is similar to the Second Doctor reluctantly becoming a CIA agent in the "Season 6B" era.