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It is known that $H^2(D_{2n},\mathbb{Z}/2\mathbb{Z})\cong(\mathbb{Z}/2\mathbb{Z})^3$, when $n$ is even. How to describe then these 8 central extensions of $D_{2n}$ explicitly, e.g. in terms of generators and relations?

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$\langle x,y,t \mid [t,x]=[t,y]=1,t^2=1, x^2=t^i, y^2=t^j, (xy)^n=t^k \rangle$ for $i,j,k \in \{0,1\}$.

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    Thank you, Derek! Do you know the reference? – 2017-02-28
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    I don't know a specific reference, but given that we know that the cohomology group is ${\mathbb F}_2^3$ and that $D_{2n}$ has a presentation with three relators, you can just write the answer down. – 2017-02-28