  We construct a neutrino model using the flavour group $S_4\times C_4 \times
C_3\times C_2$ under the type-1 seesaw mechanism. The vacuum alignments of the
flavons in the model lead to $\text{TM}_1$ mixing with $\sin
\theta_{13}=\frac{1}{\sqrt{3}}\sin \frac{\pi}{12}$. The mixing also exhibits
$\mu\text{-}\tau$ reflection symmetry. By fitting the eigenvalues of the
effective seesaw mass matrix with the observed neutrino mass-squared
differences, we predict the individual light neutrino masses. The vacuum
alignment of the $S_4$ triplet appearing in the Majorana mass term plays a key
role in obtaining the aforementioned $\text{TM}_1$ scenario. Since the
symmetries of the flavour group are not sufficient to define this alignment, we
apply the recently proposed framework of the auxiliary group in our model.
Using this framework, the $S_4$ triplet is obtained by coupling together
several irreducible multiplets that transform under an expanded flavour group
consisting of the original flavour group as well as an auxiliary group. The
vacuum alignment of each of these multiplets is uniquely defined in terms of
its residual symmetries under the expanded flavour group. As a result, the
$S_4$ triplet constructed from these multiplets also becomes uniquely defined.
