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?

As you stated, every (finite) permutation group can be represented by matrices (take $G=S_n$ and represent the permutations by permutation matrices (the permutations act naturally on a vector space of dimension $n$ over some field $K$)). Subsequently, every finite group can be embedded in a permutation group (a Theorem of Cayley), since it acts on itself by left (or right) multiplication.
