Herein, we investigate the symmetry-protected toroidal dipole resonances and conditions of their excitation in a new type electromagnetic metamaterials. These metamaterials are all-dielectric planar periodic arrays dielectric disks disposed on substrate. The elementary building blocks array trimers which distributed hexagonal unit supercells. highest geometrical symmetry supercell is C6v. analy...

Let [Formula: see text] be a finite group and text]. In this paper, we prove that if text], then is supersolvable. particular, some new characterizations of the well-known groups are obtained. We also show does not imply supersolvability for no constant