Due to their porosity and biocompatibility, C-N based graphitic sheets are currently attracting much attention. Here we present our findings on a new structure of a g-C4N3 sheet composed of the tri-ring heptazine-like units, which is energetically more stable, more elastic and isotropic than the previously proposed structure consisting of the single-ring triazines. Dynamics and thermal stability of the new structure are confirmed using phonon spectrum calculations and molecular dynamics simulations. Based on hybrid density functional theory, we demonstrate that the tri-ring unit based g-C4N3 is a semiconductor with a small band gap, sharp optical absorption peaks and high absorption intensity. Although the new structure is nonmagnetic, ferromagnetism can be introduced and the optical absorption can be tuned by applying a small strain.