TITLE: Which finite groups have faithful complex irreducible representations?
QUESTION [24 upvotes]: Obvious necessary condition is that the center must be a cyclic group. Is it sufficient (doubt here)? If not, is there any nice characterization in terms of group structure, without appealing to representations?

REPLY [11 votes]: I thought I'd add a specific example of a finite group with cyclic centre (trivial, in fact), yet no faithful irreducible complex representation (the example is from problem 2.19 of Isaacs' Character theory of finite groups, MR460423).
The group $(C_2)^4\rtimes C_3$, where $C_n$ denotes the cyclic group of order $n$ and $C_3=\langle \sigma\rangle$ acts on $(C_2)^4=\langle \tau_1,\tau_2,\tau_3,\tau_4\rangle$ via
$$\begin{align*}
 \sigma\cdot\tau_1=\tau_2 \hspace{0.5in}&\sigma\cdot\tau_2=\tau_1\tau_2
 \newline \sigma\cdot\tau_3=\tau_4 \hspace{0.5in}&\sigma\cdot\tau_4=\tau_3\tau_4.
\end{align*}$$