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Antoine
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Let $p$ be a prime and $n\geq 1$, it is known that there are exactly two extraspecial groups of order $p^{1+2n}$, denoted by $p^{1+2n}_+$ and $p^{1+2n}_-$. They fit into central extensions $$0\to(\mathbb{Z}/p)\to E\to(\mathbb{Z}/p)^n\to 0.$$$$0\to(\mathbb{Z}/p)\to E\to(\mathbb{Z}/p)^{2n}\to 0.$$ When $p$ is odd, $E=p^{1+2n}_+$ has exponent $p$ and an explicit $2$-cocycle $\beta:V\times V\to\mathbb{Z}/p$ for $E$ (where $V=(\mathbb{Z}/p)^n$$V=(\mathbb{Z}/p)^{2n}$, seen as an $\mathbb{F}_p$-vector space) is given by $\beta(v,w)=\frac{1}{2}[v,w]$, with $$[v,w]:=\displaystyle\sum_{i=0}^n{v_iw_{n+i}-w_iv_{n+i}}.$$ Is there any explicit description for $\beta$ when $p=2$?

Let $p$ be a prime and $n\geq 1$, it is known that there are exactly two extraspecial groups of order $p^{1+2n}$, denoted by $p^{1+2n}_+$ and $p^{1+2n}_-$. They fit into central extensions $$0\to(\mathbb{Z}/p)\to E\to(\mathbb{Z}/p)^n\to 0.$$ When $p$ is odd, $E=p^{1+2n}_+$ has exponent $p$ and an explicit $2$-cocycle $\beta:V\times V\to\mathbb{Z}/p$ for $E$ (where $V=(\mathbb{Z}/p)^n$, seen as an $\mathbb{F}_p$-vector space) is given by $\beta(v,w)=\frac{1}{2}[v,w]$, with $$[v,w]:=\displaystyle\sum_{i=0}^n{v_iw_{n+i}-w_iv_{n+i}}.$$ Is there any explicit description for $\beta$ when $p=2$?

Let $p$ be a prime and $n\geq 1$, it is known that there are exactly two extraspecial groups of order $p^{1+2n}$, denoted by $p^{1+2n}_+$ and $p^{1+2n}_-$. They fit into central extensions $$0\to(\mathbb{Z}/p)\to E\to(\mathbb{Z}/p)^{2n}\to 0.$$ When $p$ is odd, $E=p^{1+2n}_+$ has exponent $p$ and an explicit $2$-cocycle $\beta:V\times V\to\mathbb{Z}/p$ for $E$ (where $V=(\mathbb{Z}/p)^{2n}$, seen as an $\mathbb{F}_p$-vector space) is given by $\beta(v,w)=\frac{1}{2}[v,w]$, with $$[v,w]:=\displaystyle\sum_{i=0}^n{v_iw_{n+i}-w_iv_{n+i}}.$$ Is there any explicit description for $\beta$ when $p=2$?

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Antoine
Antoine
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Explicit $2$-cocycle for $2^{1+2n}_+$

Let $p$ be a prime and $n\geq 1$, it is known that there are exactly two extraspecial groups of order $p^{1+2n}$, denoted by $p^{1+2n}_+$ and $p^{1+2n}_-$. They fit into central extensions $$0\to(\mathbb{Z}/p)\to E\to(\mathbb{Z}/p)^n\to 0.$$ When $p$ is odd, $E=p^{1+2n}_+$ has exponent $p$ and an explicit $2$-cocycle $\beta:V\times V\to\mathbb{Z}/p$ for $E$ (where $V=(\mathbb{Z}/p)^n$, seen as an $\mathbb{F}_p$-vector space) is given by $\beta(v,w)=\frac{1}{2}[v,w]$, with $$[v,w]:=\displaystyle\sum_{i=0}^n{v_iw_{n+i}-w_iv_{n+i}}.$$ Is there any explicit description for $\beta$ when $p=2$?