Quaternion group

Thus $ij=k$, $jk=i$, and $ki=j$, while reversing the order of any two of $i,j,k$ changes the sign.

The element $-1$ is central, every element other than $1$ and $-1$ has order $4$, and $Q_8$ is nonabelian. The displayed rules give a closed associative multiplication; equivalently, $Q_8$ is the subgroup generated by $i$ and $j$ inside the units of the Hamilton division ring of quaternions.