This work presents a detailed investigation on the stability, flame macrostructure and combustion characteristics of oxy-methane/hydrogen (H-2-CH4-O-2-CO2) flames in a premixed swirl-stabilized model gas turbine combustor. The effects of equivalence ratio, inlet Reynolds number and inlet swirl number on stability and combustion characteristics were examined experimentally and numerically. A base operation case was selected as a reference to study the effects of the different parameters with equivalence ratio of 1.0, bulk mean throat velocity of 5.2 m/s, and swirl number of 0.98. Fixed compositions of the oxidizer (30%O-2/70%CO2) and fuel (50%H-2/50%CH4) streams were considered throughout the study. Large eddy simulation (LES) technique was adopted to compute the premixed turbulent flame regime. Experimentally captured flame images were compared with the flame contours obtained from the numerical simulations, and good conformations were observed. The results showed that near-stoichiometry flames exhibit strong inner recirculation zone (IRZ) and are stabilized between two shear layers. At lower equivalence ratios, the flames are corner-stabilized, and the flow is dominated by a single primary eddy in the IRZ. The calculated strain rates predict flame quenching at equivalence ratio of 0.5, which was also confirmed experimentally. It was observed that the effect of Reynolds number on the flame shape is not significant; however, the flow is dominated by larger primary eddy with strong IRZ as the Reynolds number increases. The IRZ also gets stronger with the increase in the flow swirl number. At higher swirl numbers, the flame stabilization occurs at the inner shear layer and the outer recirculation zone (ORZ) diminishes. It was found that the chemical time scale for all cases under investigation is shorter than the Kolmogorov time scale, signifying that the chemical reactions are dominated by the laminar flame speed.