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 contravening 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 instead sought to control the universe, beginning with the local galaxy. (TV: Survivors of the Flux)

At some point in Division history, the Division was no longer based on Gallifrey, and instead operated from a Control ship beyond the boundary of the Doctor's universe, in the space between this universe and another. From here, they took the widest view on history, controlling other worlds from afar, as complete outsiders. (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 after they seized the Temple of Atropos. This was the Siege of Atropos, in which the Time Lords' structure for Time was ultimately restored. (TV: Once, Upon Time)

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) With him having been abandoned by the Doctor, the Division placed a synaptic collider in Karvanista's brain that would kill him within 3 seconds if he revealed anything about his work for the Division. (TV: The Vanquishers)

Rewriting the Doctor
Eventually, the Division thanked the Doctor for their service, then wiped their memories, (TV: The Timeless Children) storing them in a biodata module, (TV: Survivors of the Flux) the same technology used with a Chameleon Arch to store a Time Lord's identity. (TV: Human Nature / The Family of Blood, Utopia)

To Tecteun, later regarded as a leader, the Doctor was a subject for experimentation, much like the universe would become to her. (TV: Survivors of the Flux) All records within the Matrix were purged which might reveal the Doctor's original identity, although there was an allegorical account, well-hidden. (TV: The Timeless Children)

Thirteenth Doctor's discoveries
The Thirteenth Doctor first encountered the Division when she became embroiled with the Judoon hired to find her fugitive incarnation in Gloucester. Witnessing her younger self, whom she had no memory of, escape from Gat, the Doctor was left disturbed by the experience. (TV: Fugitive of the Judoon)

Digging deep into the Matrix, discovered the history of the Timeless Child including the Division. He found a fictional story of Brendan masking hidden data in the Matrix, with 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. He broadcast the story to the Thirteenth Doctor's mind, (TV: The Timeless Children) as she was battling Ashad's Cybermen. After she reached the Boundary, (TV: Ascension of the Cybermen) the Master linked it to the ruins of Gallifrey and forced the Doctor to come with him. There he used the Matrix to show her true history, including her connection to the Division. (TV: The Timeless Children)

Shortly after she learnt the truth, 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 with the aid of Jack Harkness. (TV: Revolution of the Daleks)

The Doctor sought out Karvanista for information on the organization, as he was the only surviving Division agent she could find. Karvanista refused to answer her questions and kept setting traps to get rid of her. (TV: The Halloween Apocalypse)

End plan for the universe
Now led by Tecteun, in her "Awsok" incarnation, the Division learned of the Doctor's discoveries, and decided to abandon this universe for the next one, instigating a plan to destroy the universe which had been in the works for a long time, (TV: Survivors of the Flux) finally releasing the Flux, to destroy space, and the Ravagers, Swarm and Azure, to bring an end to structural 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)

By this point, the Time Force had been released already by the Ravagers, during the Great Disruption, and hardly any planets remained, (TV: Once, Upon Time) with the ultimate end plan to leave Earth for last, the Doctor's favoured planet. (TV: Survivors of the Flux)

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 Division instead. She was considered more valuable, and was recalled to Division by the Weeping Angels. (TV: Village of the Angels) On Division Control, all the way out in the void between dimensions, Tecteun revealed herself to the Doctor, explained the Division's plans, and made sure to highlight that it was in fact the Doctor's influence on the universe which "forced" them to seek this option, as they could never account for her actions, nor her power to inspire others. (TV: Survivors of the Flux)

She tempted the Doctor with the biodata module containing her stolen memories, but the Doctor did not waver in her intentions, vowing to destroy the Division and save the universe. Their confrontation was interrupted by Swarm and Azure, who had infiltrated the base, creating a psycho-temporal bridge to finally reach them. Having been pawns in her game, the Ravagers killed Tecteun, out of revenge for their capture and imprisonment, thus leaving the Division in a power vacuum. (TV: Survivors of the Flux)

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.