  In this paper, we study the integral Cayley graphs over a non-abelian group
$U_{6n}=\langle a,b\mid a^{2n}=b^3=1, a^{-1}ba=b^{-1}\rangle$ of order $6n$. We
give a necessary and sufficient condition for the integrality of Cayley graphs
over $U_{6n}$. We also study relationships between the integrality of Cayley
graphs over $U_{6n}$ and the Boolean algebra of cyclic groups. As applications,
we construct some infinite families of connected integral Cayley graphs over
$U_{6n}$.
