The successful operation of the LHC provides a great opportunity to study the processes where heavy baryons are involved. In this work we mainly study the weak transitions of AE b ! AE c . Assuming the reasonable quark-diquark structure, where the two light quarks constitute an axial vector, we calculate the widths of semileptonic decay AE b ! AE c e e and nonleptonic decay modes AE b ! AE c þ M (light mesons) in terms of the light-front quark model. We first construct the vertex function for the concerned baryons and then deduce the form factors which are related to two Isgur-Wise functions for the AE b ! AE c transition under the heavy quark limit. Our numerical results indicate that ÀðAE b ! AE c e e Þ is about 1:60 AE 0:28 Â 10 10 s À1 and ÀðAE b ! AE c þ MÞ is slightly below 1 Â 10 10 s À1 , which may be accessed at the LHCb detector. By the flavor SU(3) symmetry, we estimate the rates of b ! c . We suggest measuring weak decays of b ! c , because b does not decay via strong interactions, creating an obvious advantage.