The group A_4 consists of the identity, eight 3-cycles, and three products of two transpositions, namely (12)(34), (13)(24) and (14)(23). These three products of transpositions, together with the identity, form a normal subgroup isomorphic to . The 3-cycles have order 3, of course, so if you take one of them, say (234), it generates a subgroup isomorphic to , which acts by conjugation on the normal subgroup. Now all you have to do is to check that the whole group can be identified with the semidirect product of the normal subgroup by that action of .