CyberFaction

The Faction was a particular social group of Cyberman. They stood in contrast to Mondans. Both groups, however, originated on the planet Mondas, and were, in a broader sense, "Mondasian".

Whereas the Mondans wished to retain some connection to their biological form, the Faction saw greater logic in fully embracing cybernetics. They thus retained a smaller percentage of their organic systems than Mondans.

The Faction were also opposed to the plan of the Mondans to pilot Mondas back to Earth in order to rob their twin planet of its energy. When that plan failed, and Mondas exploded, (DW: The Tenth Planet) the Faction were the only group of Cybermen left in existence. They thus emerged as the "victorious" branch of the Cyber race. (NA: Iceberg)

Throughout his lives, the Doctor therefore mostly encountered descendants of the Faction. However, he, nor anyone else he encountered, regularly used the term "the Faction", suggesting that it was mostly a term used by Cybermen themselves.

Behind the scenes
The term "the Faction" has its origins in the non-narrative book, Doctor Who: Cybermen, by David Banks. Basically an attempt to invent terms to explain the costumes that Banks and his fellow Cyber performers were forced to wear over the years, the terms did not garner much attention amongst DWU authors, and weren't carried forward in narratvies.

Even Banks himself failed to use his own term for this sect, CyberFaction, when given the opportunity to do so in the novel Iceberg. Instead, he chose to go with the term "the Faction", which is what we use for this article. Even so, no one other than Banks has spoken of "the Faction" in DWU fiction.

Other "facts" from Banks' encyclopaedic Doctor Who: Cybermen included:
 * After the destruction of Mondas, parts of the Faction became CyberNomads and left Sol to explore Mutter's Spiral.
 * After the attack on the Gravitron, the Faction abandoned the Sol system and journey into the galaxy to colonise a suitable planet. They eventually found one, Telos, which they colonised in 2175, resulting in CyberTelosians.

Neither of these events were related in a narrative source, though, so this wiki's canon policy means that they remain ignored — except as a curiosity for this "behind the scenes" section.