Group Actions · Hard problem

Let $G=S_3\times S_2$ (order $6\cdot2=12$) act on $X=\{1,2,3\}\times\{A,B\}$ by $(\sigma,\tau)\cdot(x,c)=(\sigma(x),\tau(c))$. Find $|\operatorname{Stab}((1,A))|$.