You're forgiven, they are a cult band (and I don't mean that negatively):
http://en.wikipedia.org/wiki/Big_StarMojo magazine had an article on them a few years back where they were lauded as one of the Great White Hopes of American rock in the dark days of stadium rock that out of sheer misfortune never made it as big as it should have. Heralded as inventors of power pop (and frequently name-checked by people such as Paul Westerberg or Michael Stipe) - that form of music that never really had an audience in Europe where we prefer either pure pop or pure rock. I've seen Cheap Trick flounder twice on German open air rock concerts where they were - squeezed in between the likes of Thunder, Status Quo and Deep Purple - perceived by large parts of the audience as simply odd. People didn't know what to make of them and only "I want you to want me" (never their best song and a banal tune in my ears) elicited any audience reaction. When they played a Beatles number, audience reaction was outright hostile, like "don't you dare do that!".
They are one of the blank spaces in my CD collection, I will now order the box set that came out in 2009.