# An exceptional isomorphism

We will construct the exceptional isomorphism $S_6\cong Sp_4(\mathbb{F}_2)$.

The group $S_6$ acts on $\mathbb{F}_2^6$ preserving the usual pairing $\langle e_i,e_j\rangle=\delta_{ij}$ where the $e_i$ are the usual basis vectors.

There is an invariant line $L$, the span of $\sum_i e_i$ and an invariant hyperplane $H=\{\sum_i a_ie_i|\sum_i a_i=0\}$. Let $V=H/L$. $S_6$ acts on $V$.

Since $L$ is the radical of the pairing $\langle \cdot,\cdot\rangle$ on $H$, the pairing $\langle \cdot,\cdot\rangle$ descends to a non-degenerate bilinear pairing on $V$. As it is symmetric and we are in characteristic 2, it is automatically skew-symmetric.

The $S_6$-action preserves this pairing, hence we get our desired homomorphism from $S_6$ to $Sp_4(\mathbb{F}_2)$.

To check injectivity, it suffices to show that $(12)$ is not in the kernel, since we know all normal subgroups of $S_6$. Surjectivity then follows by a counting argument, so we get our desired isomorphism.