For studying symmetries of certain PDEs, it would be convenient if a certain type of group existed.

I am looking for a closed subgroup $G$ of the the orthogonal group $O(n)$ along with a continuous homomorphism $\phi : G\to \mathbb{Z}_2$ such that
1. If $x\in\mathbb{R}^n,\, x\ne 0$ then $\operatorname{dim}(Gx) > 0$.
2. $\phi : G\to \mathbb{Z}_2$ is surjective.
3. There exists $x\in \mathbb{R^n}$ such that $\{g\in G : gx = x\}\subseteq \operatorname{ker}\phi$.

I get the feeling such a group does not exist. As I am not great with groups/geometry, I was hoping someone could help lead me in the right direction (or direct me towards appropriate references).