I have to prove that there does not exist a surjective group algebra homomorphism from $FS_5$(the group algebra of the symmetric grpoup, $S_5$, over the field $F$) to $M_6(F)$, where $F$ is the field $\mathbb{Z}_2$ and $M_6(F)$ denotes the matrix algebra of $6\times 6$ matrices over the field $F$.
I have no idea how to prove it exactly. I am thinking which matrix doesn’t comes in range if particular map is defined. The dimension of domain algebra also bigger one. I already have link of the problem Artin-Wedderburn decomposition of $\mathbb{F}_2[S_5]/J$. But I do not know representation theory. Please give me a suggestion that does not use representation theory. Thanks.