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In order to induce a family of mixing patterns of leptons which accommodatethe experimental data with a simple mathematical construct, we construct anovel object from the hybrid of two elements of a finite group with a parameter$\theta$. This construct is an element of a mathematical structure calledgroup-algebra. It could reduce to a generator of a cyclic group if$\theta/2\pi$ is a rational number. We discuss a specific example on the baseof the group $S_{4}$. This example shows that infinite cyclic groups could givethe viable mixing patterns for Dirac neutrinos.

A novel mathematical construct for the family of leptonic mixing patterns