Diatomic molecule BeC has a complex electronic structure with a large number of low-lying excited states that are all strongly bound electronic states. Thus, the BeC molecule has the abundant spectral information. In this work, the potential energy curves and wavefunctions of \rmX^3 \textΣ ^ - , \rmA^3 \textΠ, \rmb^1 \textΔ , \rmc^1 \textΠ and \rmd^1\textΣ ^ + states of the BeC molecule are calculated by using the internally contracted multi-reference configuration interaction (MRCI) approach, which is based on the use of a dynamically weighted complete active space self-consistent field (DW-CASSCF) procedure. To improve the reliability and accuracy of calculation, the scalar relativistic corrections and the extrapolation of potential energy to the complete basis set limit are taken into account. On the basis of the calculated potential energy curves and wavefunctions, the spectroscopic constants (T_e, R_e, \omega _\rme, \omega _\rmex_\rme, \omega _\rmey_\rme, B_e, \alpha _\rme, and D_e) and permanent dipole moments of those states are determined, the results of which are in good agreement with the existing available experimental and theoretical values. The obtained permanent dipole moments indicate that the electrons transfer from Be to C and the polarity for molecule is \rmB\rme^\textδ + \rmC^\textδ - . The transition properties of the spin-allowed \rmA^3 \textΠ− \rmX^3 \textΣ ^ - , \rmc^1 \textΠ− \rmb^1 \textΔ , \rmc^1 \textΠ− \rmd^1\textΣ ^ + transitions are predicted, including the transition dipole moments, Franck-Condon factors, and radiative lifetimes. The radiative lifetimes for the \rmA^3 \textΠ− \rmX^3 \textΣ ^ - transitions are predicated to be at a \textµ\rm s level, and the good agreement with previous theoretical values is found. Radiative lifetimes for \rmc^1 \textΠ− \rmb^1 \textΔ and \rmc^1 \textΠ− \rmd^1\textΣ ^ + transitions are also evaluated at the levels of \textµ\rm s and ms, respectively. The PEC for the ground state is fitted into accurate analytical potential energy functions by using the extended-Rydberg potential function.