Hypergeometric $\tau$-Functions of the $q$-Painlev\'e System of Type $E_7^{(1)}$

We present the $\tau$-functions for the hypergeometric solutions to the $q$-Painlev'e system of type $E_7^{(1)}$ in a determinant formula whose entries are given by the basic hypergeometric function ${}_8{W}_7$. By using the $W(D_5)$ symmetry of the function ${}_8{W}_7$, we construct a set of twelve solutions and describe the action of $\tilde{W}(D_6^{(1)})$ on the set.