Unit Group of the Group Algebra FqGL(2,7)

In this paper, we consider the general linear group $GL(2, 7)$ of 2 × 2 invertible matrices over the finite field of order $7$ and compute the unit group of the semisimple group algebra FqGL(2,7), where Fq is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of FqGL(2,7), which is determined by first computing the Wedderburn decomposition of the group algebra Fq(PSL(3, 2) C₂). Here $PSL(3,2)$ is the projective special linear group of degree 3 over a finite field of 2 elements.