The Beta Geminorum system (or the Pollux system) was a star system in the Alpha Quadrant. Its primary Beta Geminorum was orbited by at least five planets. The closest starbase was Starbase 12. (TOS: "Who Mourns for Adonais?", Star Trek VI: The Undiscovered Country, production art)
When surveying Pollux V and the rest of the Pollux system in 2267, the USS Enterprise detected a statistically unexpected absence of intelligent life at first. However, the ships's crew soon encountered Apollo, the last member of his species of ancient intelligent humanoids, on Pollux IV. (TOS: "Who Mourns for Adonais?")
The location of Beta Geminorum in the Milky Way Galaxy was depicted on a Federation star chart in 2293. The Beta Geminorum system also appeared on a starchart Data and Picard were studying in Stellar Cartography in 2371. (Star Trek VI: The Undiscovered Country, production art;Star Trek Generations, display graphic)
Michael Okuda, who created the star chart The Explored Galaxy, depicted Beta Geminorum and Pollux as two distinct stars, even though they were established to be synonymous in "Who Mourns for Adonais?". In the chart, both Pollux and Beta Geminorum were located between the Tholian Assembly and Romulus. Both were identified in DS9: "Call to Arms" as being located in the Alpha Quadrant.
According to Star Trek: Star Charts (pgs. 60, 62), the Beta Geminorum system was located in the Beta Quadrant. Beta Geminorum was a K-class star with a magnitude of +3, which was ten times brighter than Sol. In the mid-22nd century, this system was located in non-aligned space. In the late 24th century, this system was located in or near Federation space.
Pollux V was also featured in the video game Star Trek: 25th Anniversary.