  We study problems relating to the D(2)-Problem for metacyclic groups of type
$G(p,p-1)$ where $p$ is an odd prime. Specifically we build on Nadim's thesis
\cite{Jamil}, which showed that the $\mathbb{Z}[G(5,4)]$-module $\mathbb{Z}$
admits a diagonal resolution and a minimal representative for the third syzygy
$\Omega_3(\mathbb{Z})$ is $R(2)\oplus[y-1)$. Motivated by this result, we show
that the $\mathbb{Z}[G(p,p-1)]$-module $R(2)\oplus[y-1)$ is both full and
straight for any odd prime $p$. Given Johnson's work on the D(2)-Problem
\cite{D2}, this leads to the conclusion that $G(5,4)$ satisfies the
D(2)-property, as well as providing a sufficient condition for the
D(2)-property to hold for $G(p,p-1)$, namely the condition that
$R(2)\oplus[y-1)$ is a minimal representative for $\Omega_3(\mathbb{Z})$ over
$\mathbb{Z}[G(p,p-1)]$, which we refer to as the condition M(p). Following this
result, we prove a theorem which simplifies the calculations required to show
that the condition M(p) holds. Finally, we carry out these calculations in the
case where $p=7$ and prove that the condition M(7) holds, which is sufficient
to show that $G(7,6)$ satisfies the D(2)-property.
