A Note on the Jacobson Radical J\(\left({\mathbb{F}}_{{2}^{n}}{D}_{2m}\right)\)

The dimension of the Jacobson radical J \(\left({\mathbb{F}}_{{2}^{n}}{D}_{2m}\right)\) $\left({\mathbb{F}}_{2^n},D_{2m}\right)$ and the algebraic structure of \(\frac{{\mathbb{F}}_{{2}^{n}}{D}_{{2}^{k}3}}{J\left({\mathbb{F}}_{{2}^{n}}{D}_{{2}^{k}3}\right)}\) $\frac{{\mathbb{F}}_{2^n}{D}_{2^{k}3}}{J\left({\mathbb{F}}_{2^n},D_{2^{k}3}\right)}$ are determined. Here, D_2m represents the dihedral group of order 2m, \({\mathbb{F}}_{{2}^{n}}\) ${\mathbb{F}}_{2^n}$ represents any finite field of characteristic 2, and \({\mathbb{F}}_{{2}^{n}}\) ${\mathbb{F}}_{2^n}$ D_2m represents the group algebra of the group D_2m over the field \({\mathbb{F}}_{{2}^{n}}\) ${\mathbb{F}}_{2^n}$ .