  Starting from considering deeper relationship between conjugacy classes and
irreducible representations of a finite group $G$, we find some quite simple
$R-$matrice defined by using finite groups. This construction produces many
sets (or topological spaces) admitting braid group actions. We introduce
conceptions "extended $R-$matrix" and "generalized extended $R-$matrix"
generalizing Turaev's enhanced $R-$matrix, which can still give invariants of
oriented links. With these new frames, we show that above $R-$matrix, together
with certain commuting pairs (essentially conjugacy classes of commuting pairs
) of $G$ can give integer invariants of oriented links. We construct some group
dominating these integer invariants and prove these groups are link invariant
by themselves. Using the language of the (colored) tangle category, we extended
above invariant to invariant of links and ribbon links colored by commuting
pairs. We show given a oriented link diagram $L$, a suitable weighted sum of
above invariant on all kinds of coloring of $L$ (by conjugacy classes of $G$)
is invariant under both two types of Kirby moves, thus giving a invariant for
closed three manifolds. We define a group dominating those invariants, and
prove this group is a three manifold invariant by itself.
