This paper considers a wireless powered communication network (WPCN) with group cooperation, where two communication groups cooperate with each other via wireless power transfer and time sharing to fulfill their expected information delivering and achieve "win-win" collaboration. To explore the system performance limits, we formulate optimization problems to respectively maximize the weighted sum-rate and minimize the total consumed power. The time assignment, beamforming vector and power allocation are jointly optimized under available power and quality of service requirement constraints of both groups. For the WSR-maximization, both fixed and flexible power scenarios are investigated. As all problems are non-convex and have no known solution methods, we solve them by using proper variable substitutions and the semi-definite relaxation. We theoretically prove that our proposed solution method guarantees the global optimum for each problem. Numerical results are presented to show the system performance behaviors, which provide some useful insights for future WPCN design. It shows that in such a group cooperation-aware WPCN, optimal time assignment has the greatest effect on the system performance than other factors.