We define in three equivalent ways an action of the linear group $GL(\mathbb{R}^{d+1})$ on the natural exponential families of $\mathbb{R}^d$. The first way uses the variance function of the family, the second uses the Laplace transform, and the third defines the action on the family through a generating measure. We then use this action to define a class of multivariate cubic natural exponential families generalizing the Letac-Mora class of real cubic families. A detailed description of this class is given.