The critical behavior of a three-dimensional model describing antiferromagnetic and structural phase transitions in cubic and tetragonal crystals with 2N-component order parameters (N>1) is investigated in the two-loop approximation of renormalized perturbation theory. The derived re normalization group expansions are resummed by the Padé-Borel method, and it is established that a stable fixed point exists in the three-dimensional phase space of effective coupling constants, consistent with earlier predictions based on the ε-expansion. For the physically interesting cases N = 2 and N = 3, however, the critical exponents corresponding to this point are found to be numerically close, not to the exponents of O(2N)-symmetric systems, but to those of the three-dimensional XY model, which differ substantially from them. A comparative analysis of the results obtained by the ε-expansion technique and by the renormalization group method in three-dimensional space shows that the latter approach provides significantly higher accuracy in numerical estimates.