A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form $x^5 + b x^3 + c x + d = 0$. Later, in 1867, this was generalized to equations of degree 6 by Joubert. We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups $S_5$ and $S_6$. In case of degree 5, the classical invariant theory of binary forms of degree 5 comes into play whereas in degree 6 the existence of an outer automorphism of $S_6$ plays an essential r\^ole. Although these consequences for equations of degree 5 and 6 have been cited and used many times in the literature, it seems unclear if the methods and ideas of Hermite and Joubert have really been understood.