I like the mattress example. I was going to say 3d symmetry point groups (since it is visual and useful plus huge impact in chemistry and physics). But...
The Reyes answer does what I want and makes it even simpler. I would really just cover that example, maybe even make some 3x5 cards with a lump and let audience play with them. Covering that one example and the table of results is fine. And the insight that two motions can equal a third one is COOL INSIGHT. That will get them a good feeling.
On top of that just give a generic discussion of group theory to answer basic questions: what course is it taught in (what year) for math majors, when chemists/physicists cover it. and that it has uses in chem/physics (spectroscopy, molecule ID, etc.) as well as "proving you can't solve 5th order polynomials the way you do with quadratic).
DO NOT show anything like wierd Z letter thing from kccrisman answer. NO, NO, NO. NO matrices, group names. NONE OF THAT. You don't have time to cover wierd notation in 15 minutes. Just do the one mattress example and then give them orienting information about the topic AS A TOPIC (not proving anything).