Let $G$ be a finite group acting on the finite set $X$. What is the minimal dimension $n$ needed to find an injective mapping $\psi: X \to \mathbb{C}^n$ together with a representation $D: G \to GL_n(\mathbb{C})$ so that $$\forall g \in G, \forall x \in X: D(g)(\psi(x)) = \psi(g.x)$$
We always have the option $$\psi: G \to \mathbb{C}[G] \cong \mathbb{C}^{|G|}, g \mapsto g$$
together with $D$ as the left multiplication on the elements of the group algebra. So the minimal dimension needed is upper bounded by $|G|$.
There are cases in which a smaller dimension is possible. For instance, let $G=S_2=\{Id, (12)\}$ the symmetric group with two elements acting on $X := S_2$ and let $$\psi: X \to \mathbb{C}, \psi(g) = \operatorname{sgn}(g)\\ D: G \to GL_1(\mathbb{C}), D(g) = \operatorname{sgn}(g)$$ This satisfies the condition and so dimension 1 is enough.
Are there any methods for determining this minimal dimension needed to linearize a group action?