As the title suggests, I'm trying to figure out what is the Stiefel Whitney class of the following representation of $D_4=\langle a,b \mid a^2=b^2=1, bab=a^{-1}\rangle=\Bbb Z_2\langle a\rangle \oplus \Bbb Z_2\langle b \rangle$, the usual Klein group. $$ \rho \colon D_4 \to O(2)$$ $$ a \mapsto \begin{matrix} -1 & 0 \\ 0 & -1 \\ \end{matrix}$$ $$ b \mapsto \begin{matrix} 1 & 0 \\ 0 & -1 \\ \end{matrix}$$
this representation induces a map $B\rho \colon BD_{2n} \to BO(2)$, and therefore it represents a well-defined 2-vector bundle. So it makes sense to compute the S-W classes of it. There is a way to prove that the first S-W class is $b \in H^1(D_{2n};\Bbb Z_2)=\hom(D_{2n};\Bbb Z_2)$, but I'm not aware of methods for computing the second S-W class.
I'm not very expert in representation theory (not at all honestly), I'm a topologist (well, I'm studying topology) that's why the question.