Abstract: A character (i.e. matrix trace of a representation) of symmetric group S(n) depends on a permutation through its cyclic structure. In the talk we will discuss block characters which depend only on the number of cycles in a permutation. We will describe the structure of the set of block characters. These characters turn out to be related both to the representation theory as they arise in the decomposition of the so-called coinvariant algebra representation of S(n); and to the combinatorics, in particular to the Eulerian numbers, which count the permutations with a given numbers of descents. We will also discuss the counterparts of the block characters for the infinite symmetric group and their relations to the representation theory of the full linear groups GL(n,q) over finite field.The talk is based on the joint work with A.V. Gnedin.