  Percolation, a paradigmatic geometric system in various branches of physical
sciences, is known to possess logarithmic factors in its correlators. Starting
from its definition, as the $Q\rightarrow1$ limit of the $Q$-state Potts model
with $S_Q$ symmetry, in terms of geometrical clusters, its operator content as
$N$-cluster observables has been classified. We extensively simulate critical
bond percolation in two and three dimensions and determine with high precision
the $N$-cluster exponents and non-scalar features up to $N \! =\! 4$ (2D) and
$N \! =\! 3$ (3D). The results are in excellent agreement with the predicted
exact values in 2D, while such families of critical exponents have not been
reported in 3D, to our knowledge. Finally, we demonstrate the validity of
predictions about the logarithmic structure between the energy and two-cluster
operators in 3D.
