In finite simple groups by Wilson, he constructs the triple cover of $A_6$ by considering the action of the subgroup of $A_6$ preserving the partition $\{12, 34, 56\}$ on the two vectors $(0, 0, 1, 1, 1, 1)$ and $(0, 1, 0, 1, \omega, \omega^*)$ where $\omega$ is a cube root of unity
He then adjoins further elements permuting the set of vectors generated, and shows that the resulting group is a triple cover of $A_6$
What is the motivation for this, and why is it exceptional?
I can somewhat see how to motivate it as we want to basically make “copies” of $A_6$ (corresponding to each cubic root of unity) lie over $A_6$, but this doesn’t explain to me
(A) why there aren’t higher covers of $A_6$
(B) why there aren’t higher covers than double of $A_n$ for $n > 7$
Similarly (for basically the same reasons) the triple cover of $A_7$ by extending the set of vectors by $(2, 0, 0, 0, 0, 0)$ and considering the symmetries of the $7$ orthogonal sets of $6$ vectors doesn’t really seem motivated to me