  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 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 role.
