= Solution
<Conjugacy classes> in the <symmetric group> are specified by <cycle type>, equivalently a <partition of an integer>[partition] of $4$. If $m_j$ counts cycles of length $j$, the number of <permutations> of that <cycle type> is $4!/\prod_j(j^{m_j}m_j!)$: cyclic rotation of each cycle and interchange of equal-length cycles do not change the <permutation>. This gives
$$
\begin{array}{c|ccccc}
\text{partition}&1^4&2\,1^2&2^2&3\,1&4\\\hline
\text{representative}&e&(12)&(12)(34)&(123)&(1234)\\
\text{class size}&1&6&3&8&6
\end{array}
$$
The sizes sum to $24$. Since the number of complex <irreducible representations> of a <finite group> equals its number of <conjugacy classes>, $S_4$ has $\boxed{5}$ inequivalent complex <irreducible representations>.
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