TITLE: Can we have many 1-dimensional rep, and very few high dimensional reps in a finite group?
QUESTION [7 upvotes]: Let $G$ be a finite group. Is it possible that there are "many" one-dimensional representations, and "very few" high-dimensional irreducible representations?
Originally, I thought it's impossible for the following reason. If $\chi_1, \chi_2$ are two one-dimensional representations such that $\chi_2 \chi_1^{-1} \not= 1$, and let $\rho$ be a high-dimensional irreducible representation. Then $\chi_1 \rho$ and $\chi_2 \rho$ are two nonequivalent irreps. When I tried to prove this formally, I feel it's not the case. 
Also, I find a finite group $G$ which has $2^{k-1}$ one-dimensional irreps, and two $2^{k-1}$-dimensoinal irreps, which is defined as follows. Let $G$ be generated by $g_1, g_2, \ldots, g_{2k-1}, -1$ such that $g_i^2 = -1$ and $g_i g_j = -g_j g_i$.
It seems that I already see the answer. But I hope to see some better explanation about it.

REPLY [6 votes]: There is another way to produce such examples. Take $V$ to be an $n$ dimensional ${\mathbb F}_2$ vector space, let $V^*$ be its dual and consider the Heisenberg group $H$ of $(n+1)\times (n+1)$ matrices with ${\mathbb F}_2$ entries, which consist of unipotent upper triangular  matrices, all of whose upper off diagonal entries are zero except those in the first row or the $n+1)$-th column. Thus we have an exact sequence 
$$1 \rightarrow {\mathbb F}_2 \rightarrow H \rightarrow V\oplus V^* \rightarrow 1. $$    It is easily seen that there is one irreducible representation $\rho$ of the Heisenberg group $H$,  of dimension $2^n$ where the centre ${\mathbb F}_2$ acts non-trivially. All the other irreducible representations  are one dimensional (and are characters of the quotient group $V\oplus V^*$.