  In the first part of the paper, we define the concept of a $G$-table of a
$G$-(co)algebra and we compute the $G$-table of some $G$-(co)algebras (here a
$G$-algebra is an algebra on which $G$ acts, semisimply, by algebra
automorphisms). The $G$-table of a $G$-(co)algebra $A$ is a set of scalars that
provides very precise and concise information about both the algebra structure
and the $G$-module structure of $A$. In particular, the ordinary multiplication
table of $A$ can be derived from the $G$-table of $A$. From the $G$-table of a
$G$-algebra $A$ we define a plain algebra $P(A)$ associated to it and we
present some basic functoriality results about $P$. Obtaining the $G$-table of
a given $G$-algebra $A$ requires a considerable amount of work but, the result,
is a very powerful tool as shown in the second part of the paper. Here we
compute the $SL(2)$-tables of the Poisson algebra structure of the even-degree
part of the cohomology associated to the cotangent bundle of the 3-dimensional
Heisenberg Lie group with Lie algebra $h$, that is
$H_E(h)=H_E^{\bullet}(h,\bigwedge^{\bullet}h)$. This Poisson $SL(2)$-algebra
has dimension 18. From these $SL(2)$-tables we deduce that the underlying Lie
algebra of $H_E(h)$ is isomorphic to $gl(3)\ltimes gl(3)_{ab}$ with the first
factor acting on the second (abelian) one by the adjoint representation. We
find it remarkable that the Lie algebra structure on $H_{E}(h)$ contains a
semisimple Lie subalgebra (in this case $sl(3)$) strictly larger than the Levi
factor of $\text{Der}(h)$, which in this case is $sl(2)\subset H^{1}(h,h)$.
This means that the Levi factor of the Lie algebra $H_{E}(h)$ has nontrivial
elements outside $H^{1}(h,h)$. Finally, this leads us to find a family of
commutative Poisson algebras whose underlying Lie structure is $gl(n)\ltimes
gl(n)_{ab}$ (arbitrary $n$) such that, for $n=3$, is isomorphic to $H_E(h)$.
