The statistical transformation models are a class of mathematical models in statistics with Lie group symmetries. Their information geometry is of much interest. This work deals with the geodesic flow arising from the statistical transformation model for the family of multivariate normal distributions on Euclidean spaces with the symmetry by the semi-direct product Lie group \(GL_+(n,\mathbb {R})\ltimes \mathbb {R}^n\) . The Fisher-Rao metric and the equation of the geodesic flow are explicitly deduced and their geometric properties are discussed. Particularly, a subriemannian structure is related with the geodesic flow.