What is the smallest group which is not a complex reflection group?
Many well known families of finite groups are complex reflection groups https://en.wikipedia.org/wiki/Complex_reflection_group For example: finite abelian groups, dihedral groups, symmetric groups.
I think $ A_5 $ is not a complex reflection group. But I imagine there are smaller examples.
Best Answer
Suppose that $\rho: Q_8\to U(n)$ is a faithful representation whose image is a reflection subgroup. Let $g_1,...,g_k$ denote the generators of $Q_8$ which map to complex reflections. Then at least two of these, say, $g_1, g_2$ have to be noncentral and not $\pm 1$ of each other. But then $g_1, g_2$ already generate $Q_8$. Let $H_1, H_2\subset {\mathbb C}^n$ be hyperplanes fixed by $\rho(g_1), \rho(g_2)$. Their intersection is fixed by the entire $\rho(Q_8)$ elementwise. Set $H:=H_1\cap H_2$. Then $\rho(Q_8)$ preserves the orthogonal complement $V=H^\perp$ and acts on it faithfully; it follows that $V$ is necessarily 2-dimensional. It has to be irreducible since $Q_8$ is nonabelian. Now, we are in business since, as you observed, there is, up to an isomorphism, unique faithful 2-dimensional representation of $Q_8$ and it is not a reflection representation.