Computing an explicit 2-cocycle

Question

$H^2(S_3, \mathbb{ℤ}) = \mathbb{ℤ}/2\mathbb{ℤ}$, so there is a unique non-split central extension of $S_3$ by $\mathbb{ℤ}$. How could I go about actually finding this extension (or a 2-cocycle for it)?

Answer 1

Consider the action of $\mathbf{Z}$ on $\mathbf{Z}/3\mathbf{Z}$ given by $n\cdot m=(-1)^nm$. Form the corresponding semidirect product $G=(\mathbf{Z}/3\mathbf{Z})\rtimes_{\pm}\mathbf{Z}$.

This is the desired group. Its center is the infinite cyclic subgroup $\{0\}\rtimes(2\mathbf{Z})$, and the quotient $(\mathbf{Z}/3\mathbf{Z})\rtimes_{\pm}(\mathbf{Z}/2\mathbf{Z})$ is isomorphic to $S_3$. This is a non-split extension, because $G$ has no element of order 2.