A concrete definition of generalized quaternion group?

Question

I just learned that $$Q_{2^{n+1}}=\langle a,b\mid a^{2^n}=1,\ a^{2^{n-1}}=b^2,\ b^{-1}ab=a^{-1}\rangle$$ is called a generalized quaternion group. But is there a more concrete and intuitive way to define this, say, for $n=3$, like the standard quaternion group $$Q_8=\{1,-1,i,j,k,-i,-j,-k\}$$ when $n=2$ ?

Answer 1

How about $$Q_{16}=\{1,i,i^2,i^3,j,k,l,m,-1,-i,-i^2,-i^3,-j,-k,-l,-m\},$$ where $$k\mathrel{\mathop:}=ij,\qquad l\mathrel{\mathop:}=i^2j,\qquad m\mathrel{\mathop:}=i^3j,$$ and $$-1\mathrel{\mathop:}=i^4=j^2=k^2=l^2=m^2$$ is the unique element of order 2?