A quicker route than listing every element is to follow the standard point-group decision flow chart, starting from the shape of the molecule rather than from the symmetry elements directly.
- Is the molecule linear? No. $CH_2=C=CH_2$ has terminal $CH_2$ groups sticking out on either side of the central carbon, so the overall shape is not a straight line of atoms; the flow chart moves past the linear ($C_{\infty v}$ / $D_{\infty h}$) branch.
- Does it have two or more $C_n$ axes with $n>2$ (a high-symmetry, cubic/tetrahedral/octahedral shape)? No, allene has only one special axis (the $C=C=C$ line acting as an $S_4$ axis), so it is not one of the high symmetry point groups.
- Find the highest-order proper rotation axis $C_n$. A simple $180$ degree rotation about the $C=C=C$ axis does not by itself map the molecule onto itself (it would need the reflection too), but combining a $90$ degree rotation with a reflection through the plane perpendicular to the axis does, so the molecule's improper axis is $S_4$, and the associated proper axis nested inside it is $C_2$ along the same line.
- Are there $n$ $C_2$ axes perpendicular to the principal axis? Yes, two $C_2$ axes, one through each terminal $CH_2$ group, each perpendicular to the central $C=C=C$ axis.
- Is there a $\sigma_h$? No, because that would force the two $CH_2$ planes to lie in the same plane, and they are twisted $90$ degrees apart.
- Are there $n$ dihedral mirror planes $\sigma_d$? Yes, one plane through each $CH_2$ group, containing the central axis.
A molecule with an $S_4$ axis, two perpendicular $C_2$ axes, two $\sigma_d$ planes, and no $\sigma_h$ is placed in the $D_{2d}$ point group by the flow chart, which is the same result reached by listing elements directly.
Let's summarize:
- Allene's perpendicular $CH_2$ ends are the key structural fact; they rule out any point group that needs a $\sigma_h$ or full planarity.
- The flow chart and the direct element count agree: the answer is $D_{2d}$.
The correct option is (D), $D_{2d}$.