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

Let $p=3d+1$ be prime in which $(\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^{\times }=⟨2,3⟩$, and let $\mathrm{\Lambda }=\mathbb{Z}[C_p⋊C_3]$ be the integral group ring of the metacyclic group $G(p,3)$ of order $3p$. The syzygies ${\mathrm{\Omega }}_r(\mathbb{Z})$ are the stable classes of intermediate modules in a free $\mathrm{\Lambda }$-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.