A six-dimensional Grand Unified Theory (GUT) is presented, where the two extra dimensions are compactified on a Z_3 orbifold in order to obtain a four-dimensional theory. A detailed analysis of the orbifold boundary conditions reveals that the gauge symmetry gets broken partially and, importantly, a discrete remnant survives. This symmetry can be broken further by a Higgs-like mechanism to a Delta(54) flavour symmetry, where matter can transform either as triplet or as singlet of Delta(54). Hence, in this scenario one can explain the origin of a flavour symmetry from the breaking of a higher-dimensional GUT. Finally, it is shown that this scenario can be embedded naturally in the framework of the heterotic string. In this case, one even finds an explanation for the existence of the six-dimensional GUT group from conformal field theory and, in addition, one obtains a geometrical interpretation of the Higgs-like breaking to Delta(54).