We investigate the properties of heavily C-doped GaN grown by hydride vapor phase epitaxy using both optical experiments and hybrid density functional theory calculations. Previous work has established that carbon acceptors (CN) give rise to a yellow luminescence band near 2.2 eV along with a blue luminescence band near 2.9 eV. Photoluminescence measurements show the yellow band shifting as a function of carbon concentration, suggesting a change in the behavior of carbon species as carbon content increases. With hybrid density functional theory we calculate the electrical and optical behavior of carbon centers containing multiple carbon impurities, which may arise in heavily doped material. We compare the behavior of these complexes to the isolated centers, and find that the dicarbon donor-acceptor (CGa-CN) complex is a candidate to explain the shift in the yellow luminescence peak. Tricarbon complexes have high formation energies and modest binding energies, and also give rise to optical transitions that are inconsistent with the observed spectra. We also identify the split dicarbon interstitial on the gallium site as a low-energy species with a large binding energy that may act to compensate carbon acceptors. Local vibrational modes are calculated for carbon impurity centers, and we compare these results to recent experiments. Dicarbon and tricarbon complexes involving CGa and CN exhibit modes that are only slightly higher than the isolated species, while carbon interstitials and related complexes give rise to vibrational modes much higher than CGa and CN.