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We construct a neutrino model using the flavour group $S_4\times C_4 \timesC_3\times C_2$ under the type-1 seesaw mechanism. The vacuum alignments of theflavons 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 theeffective seesaw mass matrix with the observed neutrino mass-squareddifferences, we predict the individual light neutrino masses. The vacuumalignment of the $S_4$ triplet appearing in the Majorana mass term plays a keyrole in obtaining the aforementioned $\text{TM}_1$ scenario. Since thesymmetries of the flavour group are not sufficient to define this alignment, weapply the recently proposed framework of the auxiliary group in our model.Using this framework, the $S_4$ triplet is obtained by coupling togetherseveral irreducible multiplets that transform under an expanded flavour groupconsisting of the original flavour group as well as an auxiliary group. Thevacuum alignment of each of these multiplets is uniquely defined in terms ofits residual symmetries under the expanded flavour group. As a result, the$S_4$ triplet constructed from these multiplets also becomes uniquely defined.

$\text{TM}_1$ neutrino mixing with $\sin \theta_{13}=\frac{1}{\sqrt{3}}\sin \frac{\pi}{12}$