${\mathrm{Mn}}_{3}\mathrm{Zn}\mathrm{C}$ possesses a magnetic phase transition at ${T}_{t}=233\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ from a ferromagnetic phase to a ferrimagnetic one with a noncollinear magnetic structure. The transition is accompanied by a structural change from cubic to tetragonal. The experimentally measured x-ray magnetic circular dichroism (XMCD) at the $\mathrm{Mn}\phantom{\rule{0.2em}{0ex}}K$ edge shows a drastic change at the magnetic phase transition, which is associated with the appearance of the noncollinear magnetic structure. The electronic structure and XMCD spectra of the ${\mathrm{Mn}}_{3}\mathrm{Zn}\mathrm{C}$ were investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital band-structure method for both the high-temperature cubic and low-temperature tetragonal noncollinear phases. Densities of valence states, spin, and orbital magnetic moments are analyzed and discussed. The origin of the XMCD spectra in the ${\mathrm{Mn}}_{3}\mathrm{Zn}\mathrm{C}$ compound is examined. The calculated results are compared with the experimental data.