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A stabilizer tableau represents each Pauli generator by a binary row $[x_1\cdots x_n\mid z_1\cdots z_n]$, with one additional bit when its sign is retained. Clifford gates update these rows by arithmetic over ${\mathbb{F}}_2$.

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.