Central extensions of dihedral groups

Question

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?

Answer 1

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