@testset "Regular matrix algebra" begin
  G = small_group(8, 4)
  A = AlgGrp(FlintQQ, G)
  B, BtoA = Hecke.regular_matrix_algebra(A)

  @test dim(B) == dim(A)

  for i = 1:dim(A)
    for j = 1:dim(A)
      @test BtoA(B[i])*BtoA(B[j]) == BtoA(B[i]*B[j])
      @test (BtoA\A[i])*(BtoA\A[j]) == BtoA\(A[i]*A[j])
    end
  end
end
