separable points on n-sphere

Question

Consider a metric space $(S^n, d)$, where $S^n$ is $n$-dimensional unit sphere and $d(x, y) = \cos ^{-1}(x \cdot y)$. I need to put as much points on the sphere s.t. distance between any two points will be larger than some constant $C$. Is it true that the maximum amount of points can be put on 6-sphere (which has the biggest area)?