I learned from Magma that $Sp_4(\mathbb{F}_2)$ has an index-2 subgroup isomorphic to $A_6$. Is it possible, given a matrix $M\in Sp_4(\mathbb{F}_2)$, to detect membership in this subgroup using a polynomial in the matrix entries? If so, what is the polynomial?
Edit: Mariano points out below that actually any function from $Sp_4(\mathbb{F}_2)\subset M_4(\mathbb{F}_2)$ to $\mathbb{F}_2$ is a polynomial, so in particular the indicator of my desired subgroup is a polynomial.
But is there a nice polynomial that does the trick? Apologies for the softness of this; but I mean to avoid the polynomial that I obtain by enumerating the elements of the subgroup in question and then just constructing the polynomials that serve as the indicators for each and then summing them. I would like the polynomial to illuminate the subgroup in some way. I am eager for your thoughts about this.