jmc

Let the three dimensions of the prism (length, width, and height, although not necessarily in that order) be $x,y,z$ such that $xy=24$, $xz=32$, and $yz=48$. Then the volume of the prism is $xyz$. Multiplying our three equations together and taking the square root of both sides, we find that $xyz=\sqrt{24\cdot 32\cdot 48}=\boxed{192}$ cubic centimeters.