  A quandle is a self-distributive algebraic structure that appears in
quasi-group and knot theories. For each abelian group A and c \in A we define a
quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class
of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin
quandles. Each G(A, c) is connected but not Latin unless A has odd order. G(A,
c) is non-medial unless 3A = 0. We classify their isomorphism classes in terms
of pointed abelian groups, and study their various properties. A family of
symmetric connected quandles is constructed from Galkin quandles, and some
aspects of knot colorings by Galkin quandles are also discussed.
