  We derive the three-loop order renormalization group equations that describe
the flat phase of polymerized membranes within the modified minimal subtraction
scheme, following the pioneering one-loop order computation of Aronovitz and
Lubensky [Phys. Rev. Lett. 60, 2634 (1988)] and the recent two-loop order one
of Coquand, Mouhanna and Teber [Phys. Rev. E 101, 062104 (2020)]. We analyze
the fixed points of these equations and compute the associated field anomalous
dimension $\eta$ at three-loop order. Our results display a marked proximity
with those obtained using nonperturbative techniques and reexpanded in powers
of $\epsilon=4-D$. Moreover, the three-loop order value that we get for $\eta$
at the stable fixed point, $\eta=0.8872$, in $D=2$, is compatible with known
theoretical results and within the range of accepted numerical values.
