The quaternion group is the group of order \(8\) \[ \begin{aligned} Q_8 &= \gens{-1, i, j, k \suchthat (-1)^2 = 1,\ i^2 = j^2 = k^2 = ijk = -1} \\ &\cong \gens{x, y \suchthat x^4 = y^4,\ x^2 = y^2,\ yxy^{-1} = x^{-1}}, \end{aligned} \] where the isomorphism sends \(x\mapsto i\) and \(y\mapsto j\). Its elements are \(\pm1,\pm i,\pm j,\pm k\).
Every element of \(\theset{i,j,k}\) squares to \(-1\), and \(ijk=-1\). Products of two distinct elements of \(\theset{i,j,k}\) taken along the cycle \(i\to j\to k\to i\) are positive, \(ij=k\), \(jk=i\), \(ki=j\), and products taken against the cycle are negative, \(ji=-k\), \(kj=-i\), \(ik=-j\):