The present work investigates the calculation of S-matrix elements for six-atom reactions combining reduced-dimensional wave packet dynamics and the quantum transition-state framework. We employ the eight-dimensional (8D) model Hamiltonian developed by Palma and Clary [J. Chem. Phys. 2000, 112, 1859-1867] and reduce basis set sizes as well as the number of wave packets by exploiting space inversion and permutation symmetry. Mode-specific chemistry in the H-2 + CH3 reversible arrow H + CH4 reaction is studied with full quantum-state resolution. Results for the H + CH4 reaction are compared to full-dimensional benchmark results. Detailed state-to-state results for the H-2 + CH3 reaction are presented for the first time. Although the "loss of memory" effect dominates for total energies up to 0.6 eV, more complex patterns emerge at higher energies. The agreement between the present reduced dimensional and the accurate full-dimensional results is generally good. However, shortcomings in the reduced-dimensional model can also be noted. They are related to the description of the symmetric and asymmetric C-H stretch motion in the CH4 molecule.