Now, I am a long (long) way from being an expert on vintage Omega, but this is one I picked up a few years ago and to the best of my knowledge the following seems accurate. Of course, if any expert here has a differing view, please do add as I am still really learning about this watch. The 2777......
1953. Part of a one year order from the British RAF for an antimagnetic and chronometer certified watch for pilots.
The
dial is the so-called M.O.D dial with a fat arrow and a circled T,
designating that it was part of a batch using tritium rather than an
earlier batch using radium (which the RAF recalled because of the
radiation issue).
Caseback engraved as follows:
6645-101000- NATO code
6B/542- Aircrew Issue
2575/53- Stores
number & year of order

