This site says that a Galaxy-class Starship has a "Captain's Gig".
What is it and do we have any examples (from any episodes) of how it looks or works?
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The "Gig" is in fact the small vessel (known as the "Captain's yacht") docked underneath the ship's saucer section on the Enterprises D and E.
The one on the Enterprise-D (The 'Calypso') was never used or even referred to in the TV show, with descriptions existing solely in the various technical manuals and trading cards:
There's a brief overview in the TNG Technical Manual:
This is one of those nifty things that we may never get to see on the show. We did briefly flirt with the idea of actually using the captain's yacht in Samaritan Snare, but it was decided to use an "executive shuttlecraft" instead. Patrick Stewart informs us that the yacht is named Calypso, after Jacques Cousteau's ship. Visual effects coordinator (and Navy veteran) Ron B. Moore points out that naval tradition would probably insist the craft be called the Captain's Gig.
It seems to be a Captain's Yacht..
The captain's yacht was a large shuttle built into the design of several Federation starship designs including the Galaxy-class and Sovereign-class. It was docked to the underside of the saucer section. Although it was built mainly for diplomatic missions it was also combat capable.
Ronald B. Moore notes in the Star Trek: The Next Generation Technical Manual that naval tradition would insist on calling such a craft the Captain's gig rather than the Captain's yacht.