quantum field theory - How to arrive at the Dirac Equation from Poincaré Algebra?

For the case of Galilean group, the time translation is given by the generator $H$. Hence,

$$\mid\psi(t)\rangle\to \mid\psi(t+s)\rangle =e^{-iHs}\mid\psi(t)\rangle$$ Which immediately is the Schrödinger equation, $$\dfrac{d}{dt}\mid\psi(t)\rangle=-iH\mid\psi(t)\rangle$$ How to arrive at the Dirac Equation from the Poincaré Algebra/Group? I know it will be much more involved. I am not expecting answers that say how Dirac thought of it or trial and error nonsense.

The necessity for Dirac equation should be visible from the Poincaré Algebra/Group?. How does spin come into the picture?