An explicit embedding of the Tits group in the compact real form of $E_6$

We construct a set of $27\times 27$ unitary matrices which give an explicit embedding of the Tits group in the compact real form of the Lie group of type $E_6$. A subset gives an embedding of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(25)$ in $F_4$.