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We study an effective gauge theory whose gauge group is a semidirect product$G = G_c \rtimes \mathit{\Gamma}$ with $G_c$ and $\mathit{\Gamma}$ being aconnected Lie group and a finite group, respectively. The semidirect product isdefined through a projective homomorphism $\gamma$ (i.e., homomorphism up tothe center of $G_c$) from $\mathit{\Gamma}$ into $G_c$. The (linear)representation of $G$ is made from $\gamma$ and a projective representation of$\mathit{\Gamma}$ over $\mathbb{C}$. To be specific, we take $SU(3)_L$ as $G_c$and $\mathbb{Z}_3 \times \mathbb{Z}_3$ as $\mathit{\Gamma}$. It is noticed thatthe irreducible projective representations of $\mathit{\Gamma}$ arethree-dimensional in spite of its Abelian nature. We give a toy model on thelepton mixing which illustrates the peculiar feature of such gauge symmetry. Itis shown that under a particular vacuum alignment the tri-bimaximal mixingmatrix is reproduced.

$SU(3)_{\rm L} \rtimes (\mathbb{Z}_3 \times \mathbb{Z}_3)$ gauge symmetry and Tri-bimaximal mixing