A C Major 6 (C6) chord contains the notes C, E, A. It is built from the major 6th formula 1 - 3 - 6, with C as the root.
Why does C6 have no G in it?
Because the 6th is what defines the chord, and the 5th is the note it can most afford to lose. With both G and A present the top of the chord gets congested and the 6th stops reading as colour. Every standard open C6 voicing drops the 5th for exactly this reason — the chord is C, E and A, and it sounds fuller for the omission.