Constructions of Quandles over Groups and Rings

We present the following constructions: extension of a trivial quandle by a group with a nontrivial abelian subgroup, a quandle over a noncommutative group by an arbitrary antiautomorphism, a quandle presenting from a generalized Alexander quandle with the replacement of the automorphism by a central antiautomorphism, and a quandle over an \( n \)-dimensional module depending on \( n(n-1)-1 \) parameters with \( n\geq 2 \) in a commutative unital ring.