Let $G$ be a $\{2,3\}$-group and $|G|=2^\alpha\cdot3^\beta$. For $p\in\{2,3\}$, define $$ \nu_p(G):=\min\left\{\log_p\left(\frac{|G|}{\chi(1)}\right)_p~\bigg|~\chi\in\mathrm{Irr}(G)\right\}, $$ where $\mathrm{Irr}(G)$ is the set of all irreducible $\mathbb{C}$-chracters of $G$.

Suppose that $\nu_2(G)=1$, $\nu_3(G)=0$. We want to study this group $G$. Exspecially, we want to know the answer to:

QUESTION: Are there two numbers $M$ and $N$ such that $\alpha<M$ and $\beta<N$?