A braid-like presentation of the integral Steinberg group of type $C_2$

We show that the Steinberg group $\text{St}(C_2,\mathbb{Z})$ associated with the Lie type $C_2$ and with integer coefficients can be realized as a quotient of the braid group $B_6$ by one relation. As an application we give a new braid-like presentation of the symplectic modular group $$\text{Sp}$_4({\mathbb Z})$.