How can we decide how a group is decomposed as a direct sum of cyclic $p$-groups from the character table? Assume the group is finite abelian and that we know the complex character table.
The number of irreducible characters will tell you the order of the group. Then, for each prime $p$ dividing the order of the group, and each $r$, the number of characters of order $p^r$ gives you the number of group elements of order $p^r$. And once you know the number of elements of each order, you can get the decomposition into a direct sum of cyclic $p$-groups.