<p>
The right-hand side of the formula
$$\pi_3(SK(G,1)) \cong \ker (G\otimes G \rightarrow G, g\otimes g' \mapsto gg'g^{-1}g'^{-1})$$
involves a special case of a crossed module
$$M\otimes N \rightarrow G, m\otimes n \mapsto (\partial m)(\partial'n)(\partial m)^{-1}(\partial' n)^{-1}$$
arising from two crossed modules \(\partial\colon M\rightarrow G\) and
\(\partial'\colon N\rightarrow G\) with common target group \(G\).
Here \(\otimes\) denotes the nonabelian tensor product introduced by Ronnie Brown and Jean-Louis Loday.
</p>
<p>
The following two commands construct the crossed modules \(M\hookrightarrow G\)
and \(N\hookrightarrow G\) arising from two normal subgroups \(M\), \(N\) of the sylow 2-subgroup \(G=Syl_2(M_{24})\) of the Mathieu group \(M_{24}\), and then compute the homotopy groups \(\pi_i(M\otimes N \rightarrow G)\) for
\(i=1,2\).
</p>
<div><code>
gap> G:=SylowSubgroup(MathieuGroup(24),2);;<br>
gap> L:=NormalSubgroups(G);;<br>
gap> M:=CrossedModuleByNormalSubgroup(G,L[100]);;<br>
gap> N:=CrossedModuleByNormalSubgroup(G,L[101]);;<br>
gap> T:=NonabelianTensorProduct(M,N);;<br>
gap> StructureDescription(HomotopyGroup(T,1));<br> "((C2 x ((C2 x C2 x C2 x C2) : C2)) : C2) : C2"<br>
gap> StructureDescription(HomotopyGroup(T,2));<br> "C2 x C2 x C2 x C2 x C2 x C2 x C2 x C2"<br>
</code></div>
<p>
The above construction can be iterated. In particular, one can construct the triple tensor product \((G\otimes G)\otimes G \longrightarrow G\) for any group \(G\). The following commands compute
$$\exp((G\otimes G)\otimes G) = 4 \, ,$$
$$|(G\otimes G)\otimes G| = 1073741824 \, ,$$
for \(G\) the 23rd group of order 32 in GAP's library of small groups.
</p>
<p>
Corresponding to the nonabelian tensor product, there is a nonabelian exterior product
$$M\wedge N \rightarrow G, m\wedge n \mapsto (\partial m)(\partial'n)(\partial m)^{-1}(\partial' n)^{-1}$$
arising from two crossed modules \(\partial\colon M\rightarrow G\) and
\(\partial'\colon N\rightarrow G\) with common target group \(G\).
The exterior product is obtained from the tensor product by adding the relation
$$m \otimes n = 1\ \ {\rm whenever}\ \partial m=\partial'n\,.$$
</p>
<p>
This exterior product can be iterated. In particular, one can construct the triple exterior product \((G\wedge G)\wedge G \longrightarrow G\) for any group \(G\). The following commands compute
$$\exp((G\wedge G)\wedge G) = 4 \, ,$$
$$|(G\wedge G)\wedge G| = 1024 \, ,$$
for \(G\) the 23rd group of order 32 in GAP's library of small groups.
</p>
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