  We theoretically investigate an antiferromagnetic structure in a hexagonal
zigzag-chain compound Ce$_3$TiBi$_5$, which shows a linear magnetoelectric
effect in metals and an unfamiliar temperature dependence of magnetic
susceptibility. We find a partial magnetic order as a coexistence of a
nonmagnetic zigzag chain and staggered antiferromagnetic zigzag chains. The
latter accompany in-plane magnetic toroidal dipole moments which are
responsible for the observed behaviors. We show that magnetic frustration
arising from competing exchange interactions plays an important role to realize
such a partial magnetic order. Furthermore, we present the important spin-orbit
coupling parameters to induce the physical phenomena driven by the magnetic
toroidal dipole moment, such as the linear magnetoelectric effect and nonlinear
transport.
