The non abelian exterior square of a group is defined Non abelian exterior square of a group.. I want to compute it for Sym$(4)$. It is well known that the following sequence of groups is exact: $$1\rightarrow M(Sym(4))\rightarrow Sym(4)\wedge Sym(4) \rightarrow A_4\rightarrow 1$$ where $M(Sym(4))$ is the Schur multiplier of $Sym(4)$. From here I can guess that the order of $Sym(4)\wedge Sym(4)=24$, but not more than that. Is there any way to compute it. My question is that $\textbf{is this short exact sequence splits?}$
We have many presentations for $Sym(4)$:
$Sym(4)=\langle x_1,x_2,x_3| x_1^2=x_2^3=x_3^4=1=x_1x_2x_3\rangle=\langle x_1,x_2,x_3| x_1^2=x_2^2=x_3^2=1=(x_1x_2)^2=(x_2x_3)^3=(x_3x_1)^4\rangle $. Can we use any presentation to compute this.