Symmetry theory of the flexomagnetoelectric effect in the Bloch lines

► There are 48 magnetic point groups of the Bloch lines. ► There are 11 time-invariant and 11 time-noninvariant enantiomorphic groups of the Bloch lines. ► The soliton like Bloch lines are derived from the symmetry classification. ► The multiple states Bloch line based magnetoelectric/multiferroic memory is proposed.