The non abelian exterior square of a group is defined Non abelian exterior square of a group.. The presentation for alternating group on four symbols $\{1,2,3,4\}$ is $A_4=\langle (12)(34), (123)\rangle$. We have a well known onto homomorphism from $A_4\wedge A_4$ to $[A_4,A_4]$, by $g\wedge h \mapsto [g,h]$. In this case $(12)(34)\wedge (123)\mapsto [(12)(34), (123)]=(14)(23)$. Now I want to check that whether $(12)(34)\wedge (123)\mapsto (13)(24)$ becomes a homomorphism or not?
I am trying to do it by calculation but its very cumbersome. Is there any easy technique to check it. In fact I want to count number of homorphisms from $A_4\wedge A_4$ to $[A_4,A_4]$.