A braid-like presentation of the integral Steinberg group of type C2

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