The Unit Group of the Group Algebra\({\mathbb{F}}_{q}\left({\mathbb{Z}}_{9}\rtimes{\mathbb{Z}}_{3}\right)\)

Let p be a prime, let \({\mathbb{F}}_{q}\) ${\mathbb{F}}_q$ be a finite field with q = pⁿ elements, and let \({\mathbb{Z}}_{9}\rtimes{\mathbb{Z}}_{3}\) ${\mathbb{Z}}_9⋊{\mathbb{Z}}_3$ be the semidirect product of the groups \({\mathbb{Z}}_{9}\) ${\mathbb{Z}}_9$ and \({\mathbb{Z}}_{3}\) ${\mathbb{Z}}_3$ . The unit group \(\mathcal{U}\) $\mathcal{U}$ ( \({\mathbb{F}}_{q}\) ${\mathbb{F}}_q$ ( \({\mathbb{Z}}_{9}\rtimes{\mathbb{Z}}_{3}\) ${\mathbb{Z}}_9⋊{\mathbb{Z}}_3$ ) of the group algebra \({\mathbb{F}}_{q}\) ${\mathbb{F}}_q$ ( \({\mathbb{Z}}_{9}\rtimes{\mathbb{Z}}_{3}\) ${\mathbb{Z}}_9⋊{\mathbb{Z}}_3$ ) is completely characterized.