Clips is a binary operation between conjugacy classes of closed subgroups of a compact group,
which has been introduced to obtain the isotropy classes of a direct sum of representations
, if the isotropy classes are
known for each factor
,
. This
allows one, in particular, to compute the isotropy classes of any reducible representation,
once one knows the decomposition into irreducible factors and the symmetry classes for
the irreducible factors. In the specific case of the three dimensional orthogonal group
, clips between some
types of subgroups of
have already been calculated. However, until now, clips between type-II
and type-III subgroups were missing. Those are encountered in 3D coupled
constitutive laws. In this paper, we complete the clips tables by computing the
missing clips. As an application, we obtain 25 isotropy classes for the standard
representation on the full 3D piezoelectric law, which involves the three elasticity,
piezoelectricity and permittivity constitutive tensors.
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