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Let $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$, $n\ge 1$, be the Galilean Liealgebra over a field of characteristic zero, where $\mathfrak{h}_{n}$ is theHeisenberg Lie algebra of dimension $2n+1$, and $\mathfrak{sl}(2)$ acts on$\mathfrak{h}_{n}$ so that $\mathfrak{h}_n\simeq V(2n-1)\oplus V(0)$ as$\mathfrak{sl}(2)$-modules (here $V(k)$ denotes the irreducible$\mathfrak{sl}(2)$-module of highest weight $k$). The isomorphism classes ofuniserial $\big(\mathfrak{sl}(2)\ltimes \mathfrak{h}_n\big)$-modules are known.  In this paper we study the tensor product of two uniserial representations of$\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. Among other things, we obtain the$\mathfrak{sl}(2)$-module structure of the socle of $V\otimes W$ and wedescribe the space of intertwining operators$\text{Hom}_{\mathfrak{sl}(2)\ltimes \mathfrak{h}_n}(V,W)$, where $V$ and $W$are uniserial representations of $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. Thisarticle extends a previous work in which we obtained analogous results for theLie algebra $\mathfrak{sl}(2)\ltimes \mathfrak{a}_m$ where $\mathfrak{a}_m$ isthe abelian Lie algebra and $\mathfrak{sl}(2)$ acts so that$\mathfrak{a}_m\simeq V(m-1)$ as $\mathfrak{sl}(2)$-modules.

Tensor products and intertwining operators between two uniserial  representations of the Galilean Lie algebra $\mathfrak{sl}(2)\ltimes  \mathfrak{h}_n$