Background: The $g$ factors and lifetimes of the 2${}_{1}^{+}$ states in the stable, proton-rich Sn isotopes have been measured, but there is scant information on neutron-rich Sn isotopes.Purpose: Measurement of the $g$ factor and the lifetime of the 2${}_{1}^{+}$ state at 1.141 MeV in neutron-rich ${}^{126}$Sn (${T}_{1/2}=2.3\ifmmode\times\else\texttimes\fi{}{10}^{5}y$).Method: Coulomb excitation in inverse kinematics together with the transient field and the Doppler shift attenuation techniques were applied to a radioactive beam of ${}^{126}$Sn at the Holifield Radioactive Ion Beam Facility.Results: $g({2}_{1}^{+})=\ensuremath{-}0.25(21)$ and $\ensuremath{\tau}({2}_{1}^{+})=1.5(2)$ ps were obtained.Conclusions: The data are compared to large-scale shell-model and quasiparticle random-phase calculations. Neutrons in the ${h}_{11/2}$ and ${d}_{3/2}$ orbitals play an important role in the structure of the 2${}_{1}^{+}$ state of ${}^{126}$Sn. Challenges, limitations, and implications for such experiments at future rare isotope beam facilities are discussed.