Here's something I've thought about for approximately twelve seconds:

How useless is the group \( \langle a,b \mid a^2 = b^2 = (ab)^3 = 1 \rangle\)? It's, like, almost ℤ₃, but with halfway points.

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@christianp Some of my best friends are equilateral triangles, and they consider it a fairly useful group.

@jsiehler *wipes clean the "days since I re-invented the dihedral groups" board*

@christianp You know, it's funny, though, early this morning I was having one of those twelve-second thoughts while I was brushing my teeth and it was, more or less, "two reflections compose to make a rotation. What sort of thing "X" is such that "Two X's compose to make a reflection?"

(The answer might be that I have to wipe clean the "days since I reinvented the complex numbers" slate but I don't know.)

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