From The TimeSplitters Wiki

U-Genix is a company that was in business for at least one year (2052). It was another name for The Brotherhood of Ultra Science and later Ultra-net. It was mainly used by Crow to continue his creation of an army of TimeSplitters and for his own personal use to find the secret of immortality.

The name of the company is a pun on 'eugenics', a philosophy which seeks to improve human genes by scientific intervention(the secret goal of U-Genix).