The group has two generators, $a,b$ with identity $e$ and multiplication table: $ \begin{array}{|l|l|l|} e & a & b & ba \\ \hline a & e & ba & b \\ \hline b & ba & a & e \\ \hline ba & b & e & a \end{array} $
I've looked up the list of groups of order 4 and this isn't $\mathbb Z_4$ nor $V_4$. So it can't be a group. But I am not able to locate where the problem is.