  We study a novel six-dimensional gauge theory compactified on the
$T^2/{\mathbb Z}_3$ orbifold utilizing the diagonal embedding method. The bulk
gauge group is $G\times G\times G$, and the diagonal part $G^{\rm diag}$
remains manifest in the effective four-dimensional theory. Further spontaneous
breaking of the gauge symmetry occurs through the dynamics of the zero modes of
the extra-dimensional components of the gauge field. We apply this setup to the
$SU(5)$ grand unified theory and examine the vacuum structure determined by the
dynamics of the zero modes. The phenomenologically viable models are shown, in
which the unified symmetry $G^{\rm diag}\cong SU(5)$ is spontaneously broken
down to $SU(3)\times SU(2)\times U(1)$ at the global minima of the one-loop
effective potential for the zero modes. This spontaneous breaking provides
notable features such as a realization of the doublet-triplet splitting without
fine tuning and a prediction of light adjoint fields.
