Monte Carlo simulations using the newly proposed Wang–Landau algorithm together with the broad histogram relation are performed to study the antiferromagnetic six-state clock model on the triangular lattice, which is fully frustrated. We confirm the existence of the magnetic ordering belonging to the Kosterlitz–Thouless (KT)-type phase transition followed by the chiral ordering which occurs at slightly higher temperature. We also observe the lower temperature phase transition of KT type due to the discrete symmetry of the clock model. By using the finite-size scaling analysis, the higher KT temperature T₂ and the chiral critical temperature T_c are respectively estimated as T₂ = 0.5154(8) and T_c = 0.5194(4). The results are in favour of the double transition scenario. The lower KT temperature is estimated as T₁ = 0.496(2). Two decay exponents of KT transitions corresponding to the higher and lower temperatures are respectively estimated as η₂ = 0.25(1) and η₁ = 0.13(1), which suggests that the exponents associated with the KT transitions are universal even for the frustrated model.