I am a little confused in the interpretation of the product of groups. Here is what's written in my notes:
Given groups $G_1,G_2,...,G_n$, recall that $G_1\times G_2\times ...\times G_n=\{(g_1,g_2,...,g_n)\}\vert g_i \in G_i \}$ for all $i \in I$. More generally, for groups $G_i$ indexed $i \in I$, we have $\Pi_{i\in I}G_i=\{(g_i)_{i\in I}\vert g_i \in G_i$ for all $i\}$.
So if I let $G_1=(*,X)$ where $X=\{something\}$ and $G_2=(*,Y)$ where $Y=\{something\}$, according to the definition above, what would $G_1 \times G_2 $ yield?