The regular representation of $S_3 = \langle e, a, a^2, b, ab, a^2b \rangle$ is realised into the $6 \times 6$ matrices acting on the six dimensional vector space spanned by each element of $S_3$.
OK, I have learned that the trivial representation is coming from the one-dimensional subspace
$V_1 \colon= \langle e + a + a^2 + b + ab + a^2b \rangle$
and the sign representation
$V_2 \colon= \langle e + a + a^2 - b - ab - a^2b\rangle$.
Q. What are the explicit bases of another two irreducible sub-representations of degree two ?