  We construct a $14$-dimensional affine variety $\Sigma^{14}_{\mathbb{A}}$
with a $\rm{GL}_3\times (\mathbb{C}^*)^3$-action. We denote by
$\Sigma^{13}_{\mathbb{A}}$ the affine variety obtained from
$\Sigma^{14}_{\mathbb{A}}$ by setting one specified variable to $1$ (we refer
the precise definition to Definition 1.1 of the paper). We show that several
weighted projectivizations of $\Sigma^{13}_{\mathbb{A}}$ and
$\Sigma^{14}_{\mathbb{A}}$ produce as weighted complete intersections examples
of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to $24$
classes of the graded ring database \cite{grdb}. Except No.360 in the database,
these prime $\mathbb{Q}$-Fano threefolds have a Type I Tom projection.
Moreover, they are not weighted complete intersections of the cluster variety
of type $C_2$ introduced by Coughlan and Ducat. We also show that a partial
projectivization of $\Sigma^{14}_{\mathbb{A}}$ has a $\mathbb{P}^2\times
\mathbb{P}^2$-fibration over the affine space $\mathbb{A}^9$.
