This paper deals with the geodesic flows of the \(\alpha \) -connections arising from the statistical transformation model for the multivariate normal distributions on \(\mathbb {R}^d\) . The probability density functions are parameterized by the semi-direct product Lie group \(GL_+\left( d,\mathbb {R}\right) \ltimes \mathbb {R}^d\) . The Fisher-Rao semi-definite metric and the Amari-Chentsov cubic tensor are left-invariant tensors on the Lie group. One can then describe the geodesic flows of the \(\alpha \) -connections as left-invariant systems. It is interesting that the geodesic flow of the Fisher-Rao semi-definite metric can be formulated in terms of subriemannian geometry. In fact, the Fisher-Rao geodesic flow is a subriemannian geodesic flow for a step-two left-invariant subriemannian structure on the semi-direct product Lie group. In this paper, an explicit formula of the Amari-Chentsov cubic tensor is obtained and consequently the equation for the \(\alpha \) -geodesic flows is found concretely.　 As preliminaries of the current paper, the general framework of information geometry is briefly reviewed in relation to the Fisher-Rao metric and the Amari-Chentsov cubic tensor, particularly in the case of statistical transformation models.