central extension
A central extension is the simplest interesting kind of group extension: one where the inner piece commutes with absolutely everything. The quotient does not twist the subgroup at all — yet the extension can still be nontrivial, because even with no action the way the two pieces are glued can fail to be a direct product. These extensions are the home of phenomena like spin groups, the Heisenberg group, and projective representations.
An extension 1 -> A -> E -> Q -> 1 is central if the image of A lies in the center Z(E) of the middle group, which forces A to be abelian and the induced action of Q on A to be trivial. Because the action is trivial, such extensions are classified by H^2(Q, A) computed with trivial action. The zero class is the direct product A × Q; a nonzero class gives a group that is not a direct product despite A being central.
Central extensions of a group G by abelian groups are governed by the Schur multiplier H_2(G, Z): there is a universal central extension when G is perfect (G = [G, G]), and its kernel is precisely the Schur multiplier. In physics and Lie theory, central extensions explain why symmetries act on quantum states only up to phase — a projective representation of Q is an honest representation of a central extension of Q by the circle group, with obstruction in H^2(Q, C^*).
1 -> {±1} -> Q_8 -> Z/2Z × Z/2Z -> 1: the quaternion group Q_8 = {±1, ±i, ±j, ±k} is a central extension of the Klein four-group by its center {±1}. It is nonsplit, representing a nonzero class in H^2(Z/2Z × Z/2Z, Z/2Z).
The quaternion group as a nonsplit central extension.
Being a direct product and being a central extension are different: every direct product A × Q is a central extension, but the quaternion group Q_8 is a central extension of Z/2Z × Z/2Z by Z/2Z that is not a direct product of A with the quotient.