Creating the Cayley table for $\mathbb{Z}_2 \times S_3$

Create the Cayley table for $\mathbb{Z}_2 \times S_3$

I know that the $\mathbb{Z}_2$ is:

\begin{array}{c|cc}+& 0 & 1 \\\hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array}

And that the Cayley table of $S_3$ is

\begin{array}{c|cccccc} \cdot & e & (12) & (13) & (23) & (123) & (132) \\\hline e & e & (12) & (13) & (23) & (123) & (132) \\ (12) & (12) & e & (132) & (123) & (23) & (12) \\ (13) & (13) & (123) & e & (132) & (12) & (23) \\ (23) & (23) & (132) & (123) & e & (13) & (12) \\ (123) & (123) & (13) & (132) & (12) & (132) & e \\ (132) & (132) & (23) & (12) & (13) & e & (123) \\\end{array}

Though I do not know how to multiply a Cayley table by a Cayley table. Any help will be appreciated.