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A cage isomer of benzene with one carbon–carbon double bond, conventionally described as tricyclo[3.1.0.0(2,6)]hex-3-ene. Its ideal cage geometry belongs to the C2v molecular point group. Molecular symmetry depends on the full three-dimensional cage rather than the apparent symmetry of a planar skeletal projection.

The conjugated hydrocarbon with a five-membered ring bearing two endocyclic double bonds and one exocyclic methylene double bond. Its ideal planar structure has the C2v molecular point group.

One choice is the C2v molecular point group, $\{E,C_2(z),{\sigma }_{xz},{\sigma }_{yz}\}$. The additional operation is a second reflection, sinceA bent water molecule has this symmetry. Choose the $z$ axis along its bond-angle bisector and put the molecule in the $xz$ plane: reflection in $yz$ exchanges its two hydrogen atoms, while reflection in $xz$ fixes their positions.

The symmetric approach preserves the C2v molecular point group with its twofold axis z and mirror planes xz and yz. The molecular ring lies in xz. Classify orbitals by reflection signs $({\sigma }_{xz},{\sigma }_{yz})$: $A_1=(+,+)$, $B_1=(+,-)$, $B_2=(-,+)$, $A_2=(-,-)$. With globally defined p-orbital phases the molecular orbital correlation diagram givesWithin either two-orbital symmetry block the lower member remains correlated with the lower product member; the non-crossing rule prevents swapping their order through an ordinary same-symmetry crossing. Crossings between different irreducible representations are allowed. The crucial consequence is that the occupied carbon pi orbital correlates to a C–O-antibonding combination, rather than both initially occupied $A_1$ orbitals becoming the two product bonding orbitals.

The C2v molecular point group has four one-dimensional irreducible representations. In the operation order used above, the character table rows are $A_1=(1,1,1,1)$, $A_2=(1,1,-1,-1)$, $B_1=(1,-1,1,-1)$ and $B_2=(1,-1,-1,1)$. Their character inner products with ${\chi }_{\Gamma }=(6,0,2,0)$ giveHence $\Gamma =2A_1\oplus A_2\oplus 2B_1\oplus B_2$, whose dimensions sum to six. The labels $B_1,B_2$ refer to the source axes; swapping $x$ and $y$ swaps these names.

The fulvene structure has an in-plane twofold axis through the exocyclic double bond and the opposite part of the five-membered ring. The molecular plane and a plane perpendicular to it containing that axis are both mirrors. There is no perpendicular sixfold axis or inversion centre. Its molecular point group is $C_{2v}$, the C2v molecular point group.

The folded benzvalene cage has one twofold axis and two mirror planes containing it. The double-bonded edge distinguishes it from cages having higher rotational symmetry; it has neither an inversion centre nor a horizontal mirror perpendicular to the twofold axis. Its molecular point group is $C_{2v}$, the C2v molecular point group.

Use the trapezium's printed axes, with principal axis $z$ in its plane and $y$ perpendicular to the plane. For either site pair $(1,2)$ or $(3,4)$, the group representation has characters of a representation $(2,0,2,0)$ in the order $(E,C_2(z),\sigma (xz),\sigma (yz))$: identity and the molecular-plane mirror fix both s orbitals, while the other two operations exchange them. The character inner products give ${\Gamma }_{12}={\Gamma }_{34}=A_1\oplus B_1$ in the C2v molecular point group.