Do all Groups have a representation? I know that many kind of groups can be represented by matrices; for example: rotation groups can be represented by matrices.  Especially all elements of rotation groups can be represented by orthogonal matrices with determinant positive one. Also Permutation group can be represented by matrices.
But I need to know: 


*

*are there any kind of group which can not be represented by matrices?          

*Or is there any kind of group which does not have a representation?              
Can you show me sample?

A representation of a group $G$ is a morphism $\rho:G\rightarrow\text{GL}(V)$ where $V$ is a vector space over a field $K$.
There are always at least the following two representations:


*

*the trivial representation, where $\rho(g)=\text{id}_V$ for all $g\in G$.

*the regular representation, where $V$ is the space of $K$-valued functions on $G$ and $\rho(g)\phi(x)=\phi(xg)$ for all $\phi\in V$ and for all $x, g\in G$.

EDIT: Also note that whenever $H$ is normal in $G$ any representation of $G/H$ can be lifted to a representation of $G$ by composing with the quotient homomorphism $G\rightarrow G/H$. 
In particular if the quotient $G/H$ is abelian this gives rise to lots of representations of $G$ since abelian groups have many representations, starting with characters (i.e. $1$-dimensional representations)
