Abstract : The measure of the complete stiffness tensor of an anisotropic elastic material bring experimental difficulties when the possible symmetries are formerly unknowns. As described mechanical measurements show great complexity, we prefer the use of an ultrasonic experimentation. These are done by using the direct contact method on a 26-faces specimen. The stiffness tensor is obtained by the minimization of a functionnal based on the discrepancy between two expression of the acoustic (Christoffel's) tensor. In order to reveal the (exact or not) symmetries of the obtained stiffness tensor expressed in the specimen's axis, pole figures show the correlation between the stiffness tensor and it's symmetric by every plane of the space. These maps are revealing every possible material symmetry. An intrinsic operator determine the stiffness tensor having the chosen symmetry the nearest from the experimentally obtained stiffness tensor. The discrepancy between the two tensors and the order of the symmetry levels allow to choose the best symmetry for the studied material. Finally, damage-induced anisotropy is investigated with Kelvin's decomposition.