Multiplicative Products of Dedekind $\eta$-Functions and Group Representations

In this paper, we findall metacyclic groups ($\langle a,b\colon a^m=e,\,b^s=e,\,b^{-1}ab=a^r\rangle$), where $m=10$, $14$, $15$, $20$, $21$, $22$, such that the cusp forms associated with all elements of these groups by an exact representation are multiplicative $\eta$-products. We also consider the correspondence between multiplicative $\eta$-products and elements of finite order in $SL(5,C)$ by the adjoint representation.