Syzygy Computations and the D(2)-Problem for the Metacyclic Group G(p,3)

Let $p=3d+1$ be prime in which $(\mathbb{Z}/p\mathbb{Z})^\times=\langle 2,\,3\rangle$, and let $\Lambda=\mathbb{Z}[C_p\rtimes C_3]$ be the integral group ring of the metacyclic group $G(p,\,3)$ of order $3p$. The syzygies $\Omega_r(\mathbb{Z})$ are the stable classes of intermediate modules in a free $\Lambda$-resolution of the trivial module. We discuss explicitly the interaction of the syzygies under $-\otimes_\mathbb{Z}-$ and relate this to Wall's $\mathcal{D}(2)$-problem.