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We will only need to know the stabilizers in

The computation of such a group stabilizer is a less straightforward matter
than the computation of the Lie algebra stabilizer. Our methods are
somewhat
ad hoc, but have in common that they use the Lie algebra in two ways:
both for finding linear tranformations fixing the form **f** by exponentiating
nilpotent elements of the Lie algebra stabilizer and for finding
-invariant linear subspaces of . We use computer algebra packages
such as Mathematica and Maple to check in each case that certain matrices in
preserve the form. Knowing the dimension of from Table **1**
gives a bound for the dimension of the stabilizer. If this is attained by a
subgroup it contains at least the connected component of the full stabilizer
and group arguments involving the normalizer can be used. If not, we use the
packages to show there are no further elements fixing the form.

The stabilizer of is (as in [Z]) a finite group
isomorphic to .
It consists of all permutation matrices together with all
products **PSQ** where **P** and **Q** are permutation matrices and
S is the matrix

Note that the forms listed are all distinct. Indeed the dimensions of
the Lie algebras are distinct except for the two of dimension **12** and
the two of dimension **14**. The ones of dimension **14**, and ,
correspond to orbit dimensions **15**
and **14** and so are distinct. The Lie algebras for and
are not isomorphic. Indeed is abelian but is
not. The forms and can be also distinguished because the tori
have different eigenvalues. Alternatively, exponentiating a
nilpotent element in the Lie algebra for fixes the form but does
not for .

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