  We investigate essential dimension of finite groups over arbitrary fields and
give a systematic treatment of multihomogenization, introduced by H.Kraft,
G.Schwarz and the author. We generalize the central extension theorem of Buhler
and Reichstein and use multihomogenization to substitute and generalize the
stack-involved part of the theorem of Karpenko and Merkurjev about the
essential dimension of p-groups. One part of this paper is devoted to the study
of completely reducible faithful representations. Amongst results concerning
faithful representations of minimal dimension there is a computation of the
minimal number of irreducible components needed for a faithful representation.
