 Singer cycle

Multiplication by a generator of ${\mathbb{F}}_{q^n}^{\times }$ on the ${\mathbb{F}}_q$-space ${\mathbb{F}}_{q^n}$ gives a cyclic subgroup regular on nonzero vectors. Its normalizer contains the field Frobenius automorphism. In degree three over ${\mathbb{F}}_2$ the full normalizer has order 21.