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

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

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