It is well-known that an element of commutator subgroup may not be a commutator.
Now there is a character-theory technique to check if an element of a group is a commutator.
An element $g\in G$ is a commutator in $G$ if and only if $$\sum_{\chi} \frac{\chi(g)}{\chi(1)}\neq 0.$$
The examples of finite groups in which some element of $G'$ is not a commutator are not so easy to produce (many examples are constructed by some ad-hoc method). For such groups, it may be difficult to quickly obtain its character table with minimum tools of character theory. The above theorem appears in chapter 3 of character theory by Isaacs, and so I will consider only these chapters.
Question: Is there simple example of a finite group, for which, we can easily determine its character table, and illustrate that some element of its commutator subgroup is not a commutator.
You might want to read I. M. Isaacs: Commutators and the Commutator Subgroup. The American Mathematical Monthly. Vol. 84, No. 9 (Nov., 1977), pp. 720-722