Puzzle #8: "Group Theory"

 It's the second of the month, and you know what that means! Time for a puzzle. A few thoughts on this one below.

 

One good reason to have a puzzle blog is to showcase puzzles that I like, but don't think will ever quite be polished enough to present without commentary. There is no theme that I want to make work than this one: it has a perfect, punny, 15-letter-long revealer, and I spent a full day scouring every source I could think of to find theme answers that work. The four here are the only answers I've found that work, and even they are quite stretches: 20-Across isn't particularly well-known stateside, 29-Across is little better, and the cluing for 55-Across is a pretty awkward joke (47-Across, however, is perfect). And the fact that these, the only answers I've found that work, all fit symmetrically into a puzzle with the revealer as a central, vertical spanner?

 It's too perfect not to make a puzzle out of. But it's a puzzle that's a nightmare to fill, with all that theme content in awkward places, and with at least two theme answers that don't quite gel for me. I hope you enjoyed!