In this paper we consider the group algebra $R(C_2\times D_\infty)$. It is shown that $R(C_2\times D_\infty)$ can be represented by a $4\times 4$ block circulant matrix. It is also shown that $\mathcal{U}(\mathbb{Z}_2(C_2\times D_\infty))$ is infinitely generated.
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