 expressing it as a semidirect product) is to observe that the automorphisms of formula_18 are the trivial group, so the only possible [semi]direct product of formula_18 with itself is a direct product (which gives rise to the Klein four-group, a group that is non-isomorphic with formula_17).An example where the Schur–Zassenhaus theorem does apply is the symmetric group on 3 symbols, formula_27, which has a normal subgroup of order 3 (isomorphic with formula_28) which in turn has index 2 in formula_27 (in agreement with the theorem of Lagrange), so formula_30.Since 2 and 3 are relatively prime, the Schur–Zassenhaus theorem applies and formula_31.