This work is licensed under Creative Commons Attribution–NonCommercial International License
(CC BY-NC 4.0).
In this paper, we consider the general linear group GL(2,7)GL(2,7) of 2×22×2 invertible matrices over the finite field of order 77 and compute the unit group of the semisimple group algebra FqGL(2,7)FqGL(2,7), where FqFq is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of FqGL(2,7)FqGL(2,7), which is determined by first computing the Wedderburn decomposition of the group algebra Fq(PSL(3,2)⋊C2)Fq(PSL(3,2)⋊C2). Here PSL(3,2)PSL(3,2) is the projective special linear group of degree 3 over a finite field of 2 elements.