  We consider a four-dimensional $SU(N_c)$ gauge theory coupled to $N_f$
species of color fermions and $N_f^2$ colorless scalars. The quantum field
theory possesses a weakly interacting ultraviolet fixed point that we determine
from beta functions computed up to four-loop order in the gauge coupling, and
up to three-loop order in the Yukawa and quartic scalar couplings. The fixed
point has one relevant direction giving rise to asymptotic safety. We compute
fixed point values of dimensionless couplings together with the corresponding
scaling exponents up to the first three nontrivial orders in Veneziano
parameter $\epsilon$, both for infinite and finite number of colors $N_c$. We
also consider anomalous dimensions for fields, scalar mass squared, and a class
of dimension-three operators. Contrary to previous studies, we take into
account possible mixing of the latter and compute eigenvalues of the
corresponding matrix. Further, we investigate the size of the conformal window
in the Veneziano limit and its dependence on $N_c$.
