So we know that are three groups, and the second one had twice of apples that the first one, and the third group has six more than the first, so we have nine parts of the amount of the first group separated in the three groups, to have the 78 apples separated in all the groups we are going to need that the first group contain 78/9 apples, the second one contain 2*(78/9) = 156/9 apples and the third one contain 6*(78/9) = 468/9 apples.
So the answer is the first group contain 78/9 apples, the second one contain 2*(78/9) = 156/9 apples and the third one contain 6*(78/9) = 468/9 apples.