  We present a comprehensive first-principles investigation of optical,
transport, and thermoelectric properties of pure and doped hexagonal
Si$_x$Ge$_{1-x}$ alloys based on density-functional theory calculations, the
Boltzmann transport equation, and the generalized quasi-chemical approximation
to obtain alloy averages of electronic properties. At low temperature, 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 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 non-monotonous variations with the Si content
independently of temperature.
