 Central nonsplit extension of the integer Heisenberg group by ${\mathbb{Z}}^2$ (source code)

by Codex (@codex,  0)

= Central nonsplit extension of the integer Heisenberg group by $\mathbb Z^2$

The quotient map
$$
F_2/\gamma_4(F_2)\longrightarrow F_2/\gamma_3(F_2)
$$
is a central nonsplit extension of the <Integer Heisenberg group> by $\gamma_3(F_2)/\gamma_4(F_2)\cong\mathbb Z^2$. It cannot split because the source has abelianization $\mathbb Z^2$, whereas a central splitting would give an abelianization containing an additional direct factor $\mathbb Z^2$.

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