  We construct a $14$-dimensional affine variety $\Sigma^{14}_{\mathbb{A}}$
with a $\rm{GL}_3$- and a $(\mathbb{C}^*)^6$-actions. 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. 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 $\mathbb{A}^9$.
