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Ge-rich hexagonal SiGe alloys have recently emerged as new direct-gap semiconductors with unprecedented potential for integration of photonics on silicon. We present a comprehensive first-principles investigation of optical, transport, and thermoelectric properties of pure and doped hexagonal SixGe1−x alloys based on density-functional theory calculations, the Boltzmann transport equation, and the generalized quasichemical approximation to obtain alloy averages of electronic properties. At low temperatures, phase decomposition into the hexagonal elementary crystals is thermodynamically favored, but around and above room temperature random alloys are predicted to be stable. While hexagonal Si has an indirect band gap, the gap of hexagonal Ge is direct with very weak optical transitions at the absorption edge. The alloy band gap remains direct for a Si content below 45% and the oscillator strength of the lowest optical transitions is efficiently enhanced by alloying. The optical spectra show clear trends and both absorption edges and prominent peaks can be tuned with composition. The dependence of transport coefficients on carrier concentration and temperature is similar in cubic and hexagonal alloys. However, the latter display an anisotropic response due to the reduced hexagonal symmetry. In particular, the transport mass exhibits a significant directional dependence. Seebeck coefficients and thermoelectric power factors of n-doped alloys show nonmonotonous variations with the Si content independently of temperature.