This paper considers precoding for multi-group multicasting with a common message. The multiple antenna base station communicates with $K$ clusters, each with $L$ users. There is a common message destined to all users and a private multicast message for each cluster. We study the weighted sum rate (WSR) maximization problem for two different schemes: (i) the base station transmits the superposition of common and multicast messages, (ii) the base station concatenates the multicast message vector with the common message. We also formulate a second problem, weighted minimum mean square error (WMMSE) minimization, and prove that WSR maximization and WMMSE minimization are equivalent at the optimal solution. Inspired by the WMMSE problem, we suggest a suboptimal algorithm, based on alternating optimization. We apply this algorithm to the two transmission schemes, and understand that there is a fundamental difference between the two. We compare the results with maximal ratio transmission (MRT), and zero-forcing (ZF) precoding, and investigate the effects of the number of base station antennas, the number of groups and the number of users in a group. Finally, we study imperfect successive interference cancellation (SIC) at the receivers and show that the first transmission scheme is more robust.