A multi-antenna transmitter that conveys independent sets of common data to distinct groups of users is herein considered, a model known as physical layer multicasting to multiple co-channel groups. In the recently proposed context of per-antenna power constrained multigroup multicasting, the present work focuses on a novel system design that aims at maximizing the total achievable throughput. Towards increasing the system sum rate, the available power resources need to be allocated to well conditioned groups of users. A detailed solution to tackle the elaborate sum rate maximizing, multigroup multicast problem, under per-antenna power constraints is therefore derived. Numerical results are presented to quantify the gains of the proposed algorithm over heuristic solutions. The solution is applied to rayleigh as well as Vandermonde channels. The latter case is typically realised in uniform linear array transmitters operating in the far field, where line-of-sight conditions are realized. In this setting, a sensitivity analysis with respect to the angular separation of co-group users is included. Finally, a simple scenario providing important intuitions for the sum rate maximizing multigroup multicast solutions is elaborated.