HMMT 2024 C9

Compute the number of triples $(f, g, h)$ of permutations on $\{1,2,3,4,5\}$ such that

$$\begin{aligned} & f(g(h(x)))=h(g(f(x)))=g(x), \\ & g(h(f(x)))=f(h(g(x)))=h(x), \text { and } \\ & h(f(g(x)))=g(f(h(x)))=f(x) \end{aligned}$$

for all $x \in\{1,2,3,4,5\}$.