I am computing the cohomology of $BO(2) \times B0(3)$ and I would like to identify the Stiefiel Whitney classes of this space.
For instance, I know $H^*(BO(2);\mathbb{Z}/2)\cong \mathbb{Z}/2[w_1,w_2]$ , $H^*(BO(3);\mathbb{Z}/2)\cong \mathbb{Z}/2[w_1',w_2',w_3']$ and the cross product $H^*(BO(2);\mathbb{Z}/2) \otimes_{\mathbb{Z}/2} H^*(BO(3);\mathbb{Z}/2) \cong H^*(BO(2) \times BO(3);\mathbb{Z}/2)$. Then if $p_1:BO(2) \times B0(3) \to BO(2)$ and $p_2:BO(2) \times B0(3) \to BO(3)$ are the projections onto the first and second factors respectively, $\alpha_i=p_1^*(w_i)$, and $b_j=p_2^*(w_j')$. I have $H^1(BO(2) \times BO(3);\mathbb{Z}/2)=\langle \alpha_1, \beta_1\rangle$, but what should be the first Stiefel Whitney class in terms of $\alpha_1$ and $\beta_1$?
Thanks in advance!