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?
A finite abelian group has a faithful irreducible representation if and only if it is cyclic. The case of finite groups was solved by Gaschütz in W. Gaschütz, Endliche Gruppen mit treuen absolutirreduziblen Darstellungen. Math. Nach. 12 (1954) From Mathematical Reviews: "In the present formulation the author calls the direct product $$ S= M_1\times M_2\times\cdots\times M_t $$ of the minimal normal subgroups $M_i$ of $G$ the `base' of $ G$, and writes $S=A\times H$, where $A$ is abelian and $H$ contains no normal abelian subgroup. The condition is as follows: a finite group $G$ has a faithful irreducible representation in an algebraically closed field of characteristic zero if and only if the base $S$ (or alternatively, $A$) of $G$ is generated by a single class of conjugates in $G$. The proof is based on an elegant application of the exclusion principle." Maybe you also want to look at Bekka, Bachir; de la Harpe, Pierre, Irreducibly represented groups. Comment. Math. Helv. 83 (2008), no. 4, 847–868 where the case of infinite groups and unitary representations on Hilbert spaces was studied. One of the main results of the paper is the following:



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 $$\begin{align*} \sigma\cdot\tau_1=\tau_2 \hspace{0.5in}&\sigma\cdot\tau_2=\tau_1\tau_2 \newline \sigma\cdot\tau_3=\tau_4 \hspace{0.5in}&\sigma\cdot\tau_4=\tau_3\tau_4. \end{align*}$$ 


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groups, it's a standard fact that having a faithful irreducible representation is equivalent to having a cyclic center. I'm not sure about the general case, but it's been discussed in many books and papers. My impression is that there is no known definitive structural condition for sufficiency. – Jim Humphreys Mar 2 '11 at 16:52