T gate ($T=\mathrm{diag}(1,e^{i\pi /4})$)

The phase gate with angle $\pi /4$. It rotates the Bloch vector by $\pi /4$ around $z$, and $T^2=S$. It is not a Clifford gate: it takes the positive $X$ eigenstate to a state with vector $(1/\sqrt{2},1/\sqrt{2},0)$, outside the stabilizer octahedron.