In this paper, we present some results on algebraic complexity theory, which is a relatively new region of complexity theory. We consider the complexity of multiplication (bilinear complexity) in group algebras. In the 6-dimensional group algebra $\mathbf C(S_3)$ over the field of complex numbers with permutations of the third order as the basis, we find a bilinear algorithm for multiplication with multiplicative complexity equal to 9 (instead of trivial 36) and prove that this bound is unimprovable. We prove a series of assertions on the structure of the group algebra $\mathbf C(S_3)$, in particular, we show that the algebra $\mathbf C(S_3)$ is decomposed into a direct product of the algebra of $2\times 2$ matrices and two one-dimensional algebras. This research was supported by the Russian Foundation for Basic Research, grant 03–01–00783.