  Exploration of nontrivial magnetic states induced by strong spin-orbit
interaction is a central topic of frustrated magnetism. Extensive studies are
concentrated on rare-earth-based magnets and 4d/5d transition metal compounds,
which are mostly described by an effective spin $S_{\text{eff}} = 1/2$ for the
Kramers doublet of the lowest crystal-electric-field levels. Variety of
magnetic orderings may be greatly enhanced when magnetic dipolar moments
intertwined with multipolar degrees of freedom which are described by
higher-rank tensors and often require the magnetic ions with $S_{\text{eff}} >
1/2$. Here, our synchrotron x-ray diffraction near the Dy $L_3$ edge has
unveiled a canted antiferromagnetic ground state arising from a quasi-quartet
($S_{\text{eff}} = 3/2$) of 4f electrons in a triangular-lattice (TL)
rare-earth intermetallics DyAuGe. Magnetic moment and electric-quadrupole
moment are closely interlocked and noncollinear magnetic-dipole alignment is
induced by antiferroic electric-quadrupole (AFQ) ordering in the TL layers. The
correlation between the AFQ and canted magnetic structures is further confirmed
by phase transitions in an in-plane magnetic field. These findings offer
insights into the emergence of nontrivial magnetic states in frustrated TL
systems described beyond the $S_{\text{eff}} = 1/2$.
