jmc

For every divisor $d$ of $12$, then $12/d$ is also a divisor of $12$. Their product is $d\cdot (12/d)=12$. It follows that every divisor can be paired with another divisor of $12$ such that their product is $12=2^2\cdot 3$. There are $(2+1)(1+1)=6$ divisors of $12$: $1,2,3,4,6,12$. Thus, the product of the divisors is given by $12^{6/2}=12^3=\boxed{1728}$.