  Let $RG$ be the gruop ring of the group $G$ over ring $R$ and
$\mathscr{U}(RG)$ be its unit group. Finding the structure of the unit group of
a finite group ring is an old topic in ring theory. In, G. Tang et al: Unit
Groups of Group Algebras of Some Small Groups. Czech. Math. J. 64 (2014),
149--157, the structure of the unit group of the group ring of the non abelian
group $G$ with order $21$ over any finite field of characteristic 3 was
established. In this paper, we are going to generalize their result to any non
abelian group $G=T_{3m}$, where $T_{3m} = \langle
x,y\,|\,x^m=y^3=1,\,x^y=x^t\rangle$.
