Grand gauge-Higgs unification on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> via the diagonal embedding method

We study a novel six-dimensional gauge theory compactified on the ${T}^{2}/{\mathbb{Z}}_{3}$ orbifold utilizing diagonal embedding method. The bulk group is $G\ifmmode\times\else\texttimes\fi{}G\ifmmode\times\else\texttimes\fi{}G$, and part ${G}^{\mathrm{diag}}$ remains manifest in effective four-dimensional theory. Further spontaneous breaking of symmetry occurs through dynamics zero modes extradimensional components field. apply this setup to $SU(5)$ grand unified examine vacuum structure determined by modes. find matter contents that radiatively induce breakings ${G}^{\mathrm{diag}}\ensuremath{\cong}SU(5)$ down $SU(3)\ifmmode\times\else\texttimes\fi{}SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ at global minima one-loop potential for This provides notable features such as realization doublet-triplet splitting without fine tuning prediction light adjoint fields.