In case some of your students like physics or engineering:
One of the last chapters of James and Liebeck's textbook on Representations and Characters of Finite Groups explains how group theory (especially some basic character theory) can be used to radically simplify calculations in a type of physics problem: the vibration of a molecule. You might be interested in this as it takes a big linear algebra problem (say 15 x 15, not so large it cannot be written down on paper) and makes it a much smaller problem (say 3 x 3, something they might be more inclined to solve). We regularly assign problems that look like the first versions of these in calc 2, and some of the medium sized ones in calc 4, so perhaps they may help encourage continuity between courses.
We studied steady states of "trusses" and other aspects of bridge construction in our second undergrad linear algebra class, and I found it fascinating. I would have been very impressed that group theory could have dramatically shrunk those problems.
There are some undergraduate texts on the hydrogen atom that are basically representations of groups. Several good texts are available online from Springer, and can be used to fill out Arturo's suggestion of quantum mechanics. Again these will mesh very well with linear algebra, especially a second course.
If the students are familiar with differential equations, then exploiting simple symmetries in the equations to find more solutions is good. The clearest example of this is a DE with real equations and a complex solution; the equation is invariant under complex conjugation, so the complex conjugate of the solution will also be a solution. One can use larger symmetries, such as a rotation invariant PDE etc.