A presentation of the alternating group $A_4$ is $$ A_4 =\langle a,b\mid a^2=b^3=(ab)^3=1\rangle $$ Find two elements of $S_4$ that represent $a$ and $b$.
The representatives of $a$ and $b$ that I have to find are two elements $A$ and $B$ in $S_4$ such that
$$A^2=B^3=(AB)^3=1$$
I tried with $A=(12)(34)$ and $B=(123)=(12)(13)$ and it works, but I chose those elements quite at random and I had to do the whole calculation to check it.
Here is the full calculation of $(AB)^3$ ($A$ is denoted by $\to$ and $B$ by $\longrightarrow$) $$ \{1,2,3,4\}\to\{2,1,4,3\}\longrightarrow\{3,2,4,1\}\\\to\{4,1,3,2\}\longrightarrow\{4,2,1,3\}\\\to\{3,1,2,4\}\longrightarrow\{1,2,3,4\} $$
Is there a faster and/or clever way to find $A$ and $B$?