The operation which takes the Cayley tables of $G_1$ and $G_2$ and produces the Cayley table of $G_1\times G_2$ is sometimes called the Kronecker product, or tensor product, of matrices. The elements of $G_1\times G_2$ are ordered pairs $(g_1,g_2)$, and the group operation is coordinatewise.
For your example, you're working with a permutation representation of $S_3$, so it might be convenient to also use a representation of ${\Bbb Z}_2$ as the two permutations $(4,5)$ and $\textrm{id}=(4)(5)$. The point is that the set $\{4,5\}$ is disjoint from $\{1,2,3\}$, so the action of $S_3$ on $\{1,2,3\}$ and ${\Bbb Z}_2$ on $\{4,5\}$ together generate a (faithful) action of $S_3\times{\Bbb Z}_2$ on $\{1,2,3,4,5\}$.