  We explore a knot invariant derived from colorings of corresponding
$1$-tangles with arbitrary connected quandles. When the quandle is an abelian
extension of a certain type the invariant is equivalent to the quandle
$2$-cocycle invariant. We construct many such abelian extensions using
generalized Alexander quandles without explicitly finding $2$-cocycles. This
permits the construction of many $2$-cocycle invariants without exhibiting
explicit $2$-cocycles. We show that for connected generalized Alexander
quandles the invariant is equivalent to Eisermann's knot coloring polynomial.
Computations using this technique show that the $2$-cocycle invariant
distinguishes all of the oriented prime knots up to 11 crossings and most
oriented prime knots with 12 crosssings including classification by symmetry:
mirror images, reversals, and reversed mirrors.
