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The nitrogen pair and the four-carbon set are separately invariant under the D2h point group. Construct symmetry-adapted linear combinations within these two sets; mixing equivalent representations across the sets is left for the Hückel method calculation.

For the nitrogen $p_z$ atomic orbitals, normalized combinations areFor the carbon $p_z$ atomic orbitals, takeThese six functions are orthonormal under the zero-overlap assumption. The suffixes $x,y,xy$ describe coefficient patterns across the carbon sites; the sign acquired by each underlying $p_z$ function must also be included when assigning an irreducible representation. In particular, reflection through the molecular plane changes the sign of every function, while inversion sends ${\chi }_1$ to $-{\chi }_4$. Ignoring that orbital sign would give incorrect $g/u$ labels.

The complete $\pi$ group representation is thereforeThe diagram shows all six ring symmetry-adapted linear combinations and the resulting energy ordering used in part (iii).

Pyrazine belongs to $D_{2h}$, the D2h point group. With the printed axes, its three twofold axes are $x,y,z$; it also has an inversion centre and the three coordinate mirror planes. The nitrogen atoms occupy sites 1 and 4 on the $y$ axis. Interchanging carbon and nitrogen is not a molecular symmetry operation, so the sixfold benzene symmetry is lost.

The planar para-quinone framework has three mutually perpendicular twofold axes, inversion through its centre, and the associated mirror planes. With $x$ horizontal, $y$ along the two C=O bonds and $z$ perpendicular to the molecule, it belongs to the D2h point group, $\boxed{D_{2h}}$.

A planar six-membered aromatic ring with nitrogen at opposite ring sites. Its molecular point group is the D2h point group. The six ring $p_z$ atomic orbitals form a six-electron conjugated system; the in-plane nitrogen lone pairs are outside this $\pi$ basis.