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Let us first look at the 3-dimensional case.
Denote by U the subspace of V
spanned by the monomials that do not have z in them,
that is
The stabilizer 
in G of U induces the full 
on 
Proposition.
The orbits of 
in U are:
Use (cf. [Kraft]):
The form represents a cubic curve modulo ``cubes''.
First bring the form in Weierstraß normal form
and then look at it modulo cubes. A direct proof is given in
Appendix 1.
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