 Note that the automorphism group of formula_28 is formula_18 and the automorphism of formula_28 used in the semidirect product that gives rise to formula_27 is the non-trivial automorphism that permutes the two non-identity elements of formula_28.Furthermore, the three subgroups of order 2 in formula_27 (any of which can serve as a complement to formula_28 in formula_27) are conjugate to each other.The non-triviality of the (additional) conjugacy conclusion can be illustrated with the Klein four-group formula_40 as the non-example.