In this work we used first principles calculations in the frame of density functional theory (DFT) in order to study the structural and electronic properties of GaN doped with carbon. The computational calculations were carried out by a method based on plane waves pseudopotentials, as implemented in the Quantum Espresso code. In the wurtzite type GaN supercell the nitrogen atoms were replaced by carbon atoms (C by N) and then also the gallium atoms by carbon atoms (C by Ga). The carbon concentrations in the GaN volume was set as x=25, 50 y 75%. For each concentration x of carbon the formation energy was calculated for the substitutions C by N and CxGa. We found that it is more energetically favourable that the carbon atoms occupy the positions of the nitrogen atoms (C by N), because in all the x concentrations of carbon the formation energies were lower than that in the substitutions (C by Ga). It was found that the new compounds CxGaN1-x have higher bulk moduli. So they are very rigid. This property makes them good candidates for applications in hard coatings or devices for high power and temperatures. Analysis of the density of states show that the new CxGaN1-x ternary compound have metallic behaviour that comes essentially from the hybridization states N-p and C-p cross the Fermi level.