This is probably going to be a tedious job for someone. I apologize in advance.

I have posted a multiplication table for this group. The group is generated by two elements a, and b, both cyclic and of order 4 and 3 respectively. That is to say that all the group elements are in the form $\displaystyle a^mb^n$ where $\displaystyle a^4 = e$ and $\displaystyle b^3 = e$. The only other piece of information needed to create the table is that I have set $\displaystyle ba = a^3b$.

The problem I have is finding the center. By my definition, the center of a group G is the set of elements $\displaystyle \{ g \in G |gx = xg ~\forall x \in G \}$. I have determined the center of this group to be $\displaystyle \{ e, a^2, b^2, a^2b^2 \}$. The center is an Abelian subgroup, so the set I have labeled must be a group.

But it is not closed. In particular $\displaystyle (b^2)^2 = b$ and $\displaystyle (a^2b^2)^2 = a^3$.

What have I done wrong?

-Dan

Edit: That attachment is huge! I'll have to figure out how to reduce the size at some point.