Suppose we have $n$ points on the unit sphere,

$X = (\pmb{r}_{1}, \pmb{r}_{2}, ..., \pmb{r}_{n})$.

I am interested in knowing to what extent the rotational symmetry of $X$ is determined by the following constraint. Every triple of points is related to at least one other triple of points by some rotation about the origin,

$(\pmb{r}_{j_{1}}, \pmb{r}_{j_{2}}, \pmb{r}_{j_{3}}) = (R\pmb{r}_{l_{1}}, R\pmb{r}_{l_{2}}, R\pmb{r}_{l_{3}})$,

where $R$ is a rotation (not the identity) depending on $j_{1}, j_{2}, j_{3}$.

Any advice would be much appreciated.