For a given prime $p > 2$, what is the the smallest $n$ such that there is a finite nonabelian simple subgroup of $SO(n)$ with order divisible by $p$? It's easy to see that such an n satisfies $n \le p-1$, as $A_p$ is the rotational symmetry group of an $p-1$ simplex in $\mathbb{R}^{p-1}$, hence a subgroup of $SO(p-1)$, but occasionally one can do better: for example, the icosahedron has rotational symmetry group $A_5$, which of course sits inside $SO(3)$
The motivation for this question is precisely that icosahedral example, which feels exceptional (indeed, I believe this is the only time $A_n$ ($n>5$) sits inside $SO(n-2)$, though of course that doesn't resolve my question). I'm trying to ask when do primes appear "before their time" as the symmetries of nice geometric objects; of course, the simplicity condition is to rule out examples that are just built out of lower dimensional constructions (like fitting $C_7 \times C_{11}$ in $SO(4)$); maybe simplicity isn't the most natural condition to get at this idea of "geometrically early primes", in which case I'd be happy to hear alternatives. I might also be interested in hearing about times primes occur exceptionally in the above fashion, even if its not really a particularly low dimensional example, as long as its surprising in some way.