No onto map between group algebras $FS_5$ onto $M_6(F)$.

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.

• Can you recall what $FS_5$ is? – mathcounterexamples.net Jan 1 at 5:22
• I think the group algebra on the symmetric group $S_5$ over the field $F$. – mouthetics Jan 1 at 5:48
• @mathcounterexamples.net Yes i think i already told about group algebra... – neelkanth Jan 1 at 5:53
• If there was such a map, there would have to be an irreducible six-dimensional representation of $S_5$ over $F$. Can you determine the dimensions of the irreducible representations? – Lord Shark the Unknown Jan 1 at 6:37
• @LordSharktheUnknown I am not having knowledge of Representation theory... – neelkanth Jan 1 at 6:38