A multiple conjugation quandle and handlebody-knots

We introduce a notion of multiple conjugation quandles and give some examples of it. A multiple conjugation quandle has a mixed structure of a group and a quandle. We see that the axioms of a multiple conjugation quandle are natural from the point of view of colorings for handlebody-knots, where a handlebody-knot is a handlebody embedded in the 3-sphere. A multiple conjugation quandle is not only used to define a coloring for handlebody-knots but also to provide a universal structure to define a quandle coloring invariant in a sense.