  We will consider a two dimensional "symmetric" subfamily of the four
dimensional family of Fricke cubic surfaces. The main result is that such
symmetric cubic surfaces arise as character varieties for the exceptional group
of type G_2. Further, this symmetric family will be related to the fixed points
of the triality automorphism of Spin(8), and an example involving the finite
simple group of order 6048 inside G_2 will be considered.
