  We propose a model of fermion masses and mixings based on $SU(5)$ grand
unified theory (GUT) and a $D_{4}$ flavor symmetry. This is a highly predictive
4D $SU(5)$ GUT with a flavor symmetry that does not contain a triplet
irreducible representation. The Yukawa matrices of quarks and charged leptons
are obtained after integrating out heavy messenger fields from renormalizable
superpotentials while neutrino masses originate from the type I seesaw
mechanism. The group theoretical factors from 24- and 45-dimensional Higgs
fields lead to ratios between the Yukawa couplings in agreement with data,
while the dangerous proton decay operators are highly suppressed. By performing
a numerical fit, we find that the model captures accurately the mixing angles,
the Yukawa couplings and the $CP$ phase of the quark sector at the GUT scale.
The neutrino masses are generated at the renormalizable level with the
prediction of trimaximal mixing while an additional effective operator is
required to account for the baryon asymmetry of the universe (BAU). The model
is remarkably predictive because only the normal neutrino mass ordering and the
lower octant of the atmospheric angle are allowed while the $CP$ conserving
values of the Dirac neutrino phase $\delta_{CP}$ are excluded. Moreover, the
predicted values of the effective Majorana mass $m_{\beta \beta}$ can be tested
at future neutrinoless double beta decay experiments. An analytical and a
numerical study of the BAU via the leptogenesis mechanism is performed. We
focused on the regions of parameter space where leptogenesis from the lightest
right-handed neutrino is successfully realized. Strong correlations between the
parameters of the neutrino sector and the observed BAU are obtained.
