We have studied the room-temperature long-range antiferromagnetic order in the Fe2C monolayer using the combination of first-principles calculations and the Green\\\'s function analysis within the anisotropic Heisenberg model. The Fe2C monolayer is a semimetal with the out-of-plane antiferromagnetic order between two Fe planes. The Green\\\'s function approach within the random phase approximation is formulated to calculate the temperature-dependent antiferromagnetic magnon energy and the spin correlation function in a two-dimensional antiferromagnetic monolayer. The correlation function is used to evaluate the sublattice magnetization and study the magnetic phase transition in monolayer Fe2C. Moreover, the spin Hamiltonian and the Green\\\'s function formalism are developed to investigate the antiferromagnetic Fe2C monolayer in the presence of an external magnetic field applied along the easy axis. The H-T phase diagram shows that the antiferromagnetic to spin-flop and the spin-flop to paramagnetic phase transitions occur in low temperatures. Finally, we estimate the Néel temperature and the critical values of the magnetic field strength for these two phase transitions.

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