  We present the first implementation in the $(\beta,\gamma)$ plane of the
generator coordinate method with full triaxial angular momentum and particle
number projected wave functions using the Gogny force. Technical details about
the performance of the method and the convergence of the results both in the
symmetry restoration and the configuration mixing parts are discussed in
detail. We apply the method to the study of $^{24}$Mg, the calculated energies
of excited states as well as the transition probabilities are compared to the
available experimental data showing a good overall agreement. In addition, we
present the RVAMPIR approach which provides a good description of the ground
and gamma bands in the absence of strong mixing.
