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Point Group Finder (Molecular Symmetry)

In short: enter a SMILES or pick a classic molecule to get its point group, symmetry elements, order, chirality, polarity and character table, with every axis and mirror plane drawn in 3D. It is free, works in your web browser and needs no download.

The tool builds the 3D shape of the molecule, finds every rotation axis, mirror plane, centre of inversion and improper axis, and names the point group with its Schoenflies symbol.

Quick answer: water (H₂O) is C_2v: E, one C₂ axis and two σ_v planes, order 4. Ammonia is C_3v, methane T_d, BF₃ D_3h, SF₆ O_h, benzene D_6h and CO₂ D_∞h.

SMILES

Gallery Tolerance

How to use the point group finder

Press a gallery button such as H₂O or BF₃, choose any molecule from the list, or type a SMILES string such as O, c1ccccc1 or Cl/C=C/Cl.
Read the point group, its order, and whether the molecule is chiral and polar.
Rotate the 3D model. Axes, mirror planes, the centre of inversion and S_n axes are drawn in colour. Hover an element in the list to highlight it.
Follow the yes/no steps that lead to the group, and read its character table.

For a SMILES, MolDraw's structure engine adds the hydrogens, builds several 3D shapes, relaxes them and keeps the one with the lowest energy. Flexible molecules are judged in that shape (ethane comes out staggered). The gallery molecules use ideal, exact geometries. Small distortions are averaged out within the tolerance.

Symmetry elements and operations

A symmetry operation moves the molecule so that it looks exactly the same afterwards. A symmetry element is the point, line or plane the operation uses. One element can give several operations: a C₃ axis gives C₃ and C₃².

Symbol	Element	Operation	Example
E	Identity	Do nothing. Every molecule has it.	All molecules
C_n	Proper rotation axis	Rotate by 360°/n. The axis with the highest n is the main axis.	C₂ in H₂O, C₃ in NH₃, C₆ in benzene
σ	Mirror plane	Reflect through the plane. σ_h is at right angles to the main axis; σ_v and σ_d contain it.	σ_h in BF₃, σ_v in NH₃, σ_d in allene
i	Centre of inversion	Send every atom through the centre to the opposite side.	SF₆, CO₂, trans-1,2-dichloroethene
S_n	Improper rotation axis	Rotate by 360°/n, then reflect through the plane at right angles. S₁ = σ and S₂ = i.	S₄ in CH₄ and allene, S₆ in staggered ethane

The point group flowchart

Is the molecule linear? Yes: with a centre of inversion it is D_∞h (CO₂, H–C≡C–H); without one it is C_∞v (HCl, HCN).
Several axes with n ≥ 3? Yes: it is a high-symmetry group. Four C₃ axes and no C₄: tetrahedral, T_d (CH₄). Three C₄ axes: octahedral, O_h (SF₆). Six C₅ axes: icosahedral, I_h (C₆₀). Without mirror planes these become T, O or I.
Any rotation axis C_n? No: a mirror plane gives C_s, a centre of inversion gives C_i, nothing gives C₁.
Find the main axis C_n. Are there n C₂ axes at right angles to it?
Yes (D groups): σ_h present → D_nh; else n σ_d planes → D_nd; else D_n.
No (C and S groups): σ_h present → C_nh; else n σ_v planes → C_nv; else an S_2n axis → S_2n; else C_n.

The tool shows the path it took for each molecule under "How the group was found".

Common point groups at a glance

Group	Elements	Order	Examples
C₁	E	1	CHFClBr
C_s	E, σ	2	HOCl, CH₃OH
C₂	E, C₂	2	H₂O₂
C_2v	E, C₂, 2σ_v	4	H₂O, SO₂, CH₂Cl₂, cis-1,2-dichloroethene
C_3v	E, C₃, 3σ_v	6	NH₃, CHCl₃, PCl₃
C_2h	E, C₂, σ_h, i	4	trans-1,2-dichloroethene, trans-N₂F₂
D_2h	E, 3C₂, 3σ, i	8	Ethene, naphthalene, diborane
D_3h	E, C₃, 3C₂, σ_h, S₃, 3σ_v	12	BF₃, PCl₅, eclipsed ethane, cyclopropane
D_4h	E, C₄, 4C₂, σ_h, 4σ, i, S₄	16	XeF₄, [PtCl₄]²⁻
D_6h	E, C₆, 6C₂, σ_h, 6σ, i, S₆, S₃	24	Benzene
D_2d	E, S₄, 3C₂, 2σ_d	8	Allene
D_3d	E, C₃, 3C₂, 3σ_d, i, S₆	12	Staggered ethane, chair cyclohexane
T_d	E, 4C₃, 3C₂, 3S₄, 6σ_d	24	CH₄, CCl₄, P₄
O_h	E, 3C₄, 4C₃, 6C₂, 9σ, i, S₄, S₆	48	SF₆, cubane
C_∞v	E, C_∞, ∞σ_v	∞	HCl, HCN, OCS
D_∞h	E, C_∞, ∞C₂, ∞σ_v, σ_h, i	∞	CO₂, H₂, ethyne

Chirality, polarity and character tables

Chirality

A molecule is chiral when it has no improper axis: no mirror plane, no centre of inversion and no S_n. Only the groups C₁, C_n, D_n, T, O and I qualify. CHFClBr (C₁) and H₂O₂ (C₂) are chiral. A molecule with only an S₄ axis is not chiral, even with no mirror plane. To assign R and S to a stereocentre, use the R/S chirality finder.

Polarity

A dipole moment must point the same way after every operation. That is only possible in C₁, C_s, C_n and C_nv (including C_∞v). Water (C_2v) and ammonia (C_3v) are polar; CO₂, BF₃, CH₄, SF₆ and trans-1,2-dichloroethene are not.

Reading a character table

Each column is a class of operations, such as 2C₃ (the C₃ and C₃² rotations). Each row is an irreducible representation. Its characters show how an orbital or vibration behaves under each operation: 1 means it is unchanged, −1 means it changes sign. Mulliken labels: A and B are one-dimensional (A is symmetric about the main axis, B is not), E is two-dimensional and T three-dimensional; g and u mean symmetric or antisymmetric under inversion; ′ and ″ mean symmetric or antisymmetric under σ_h. The last two columns list where x, y, z, the rotations R_x, R_y, R_z and the quadratic functions (x², xy and so on) belong. Those columns tell you which vibrations are IR active (x, y, z) and Raman active (quadratic).

Worked examples

Water, H₂O

Bent, not linear. One C₂ axis through O. No C₂ at right angles, no σ_h, but two σ_v planes.

C_2v, order 4, polar, not chiral

Ammonia, NH₃

C₃ axis through N. No perpendicular C₂, no σ_h, three σ_v planes, each through one N–H bond.

C_3v, order 6, polar

Boron trifluoride, BF₃

C₃ axis, three C₂ axes along the B–F bonds, and the molecular plane is σ_h.

D_3h, order 12, nonpolar

Methane, CH₄

Four C₃ axes (one along each C–H bond), three S₄ axes, six σ_d planes, no centre.

T_d, order 24, nonpolar

trans- and cis-1,2-dichloroethene

trans: C₂ at right angles to the plane, the plane is σ_h, plus i. cis: C₂ in the plane and two σ_v.

trans C_2h (nonpolar); cis C_2v (polar)

Ethane: staggered and eclipsed

Both have a C₃ axis and three perpendicular C₂ axes. Staggered has i and three σ_d; eclipsed has σ_h.

Staggered D_3d; eclipsed D_3h

Allene, H₂C=C=CH₂

The two CH₂ planes are at 90°. Main axis is S₄ (with C₂), two perpendicular C₂ axes, two σ_d.

D_2d, order 8

CHFClBr

Four different atoms around carbon. No axis, no plane, no centre.

C₁, chiral and polar

Common mistakes

Working from a flat drawing. Point groups need the 3D shape. Cyclohexane is D_3d as a chair, not D_6h; staggered and eclipsed ethane differ.
Missing the S_n axis. CH₄ and allene both have S₄ axes. Without them you would pick the wrong group.
Mixing up σ_v and σ_d. In D groups, planes that bisect two perpendicular C₂ axes are σ_d.
Choosing the wrong main axis. The main axis has the highest n. In XeF₄ it is the C₄ axis at right angles to the plane, not a C₂ in the plane.
Counting elements instead of operations. The order counts operations: a C₃ axis gives two (C₃ and C₃²).
Thinking lone pairs add symmetry. Only atom positions count, but lone pairs shape the geometry. NH₃ is pyramidal (C_3v), not planar.

Want to explore a bigger molecule? Draw it in the MolDraw editor and open the Symmetry panel to play each operation in 3D.

Open MolDraw editor

Frequently asked questions

What is the point group of water?

Water is C2v. It has the identity E, one C2 axis through the oxygen and two vertical mirror planes (one is the molecular plane). The group has 4 operations, so its order is 4.

How do you find the point group of a molecule?

Use the flowchart. Is the molecule linear? Does it have several high-order axes (Td, Oh, Ih)? If not, find the highest-order rotation axis Cn. Check for n C2 axes at right angles to it (D groups), then for a horizontal mirror plane (Dnh or Cnh), then for vertical or dihedral planes (Cnv or Dnd). With no axis at all, look for a mirror plane (Cs) or a centre of inversion (Ci), else C1.

What is the difference between sigma v and sigma d?

Both are mirror planes that contain the main axis. A dihedral plane (σd) bisects the angle between two C2 axes that are perpendicular to the main axis, or lies between two σv planes. Planes that contain those C2 axes, or the outer atoms in Cnv molecules, are labelled σv.

How does symmetry tell you if a molecule is chiral?

A molecule is chiral when it has no improper axis: no mirror plane, no centre of inversion and no Sn axis. So only molecules in the groups C1, Cn, Dn, T, O and I can be chiral. CHFClBr (C1) and H2O2 (C2) are chiral; water (C2v) is not.

Which point groups can have a dipole moment?

Only C1, Cs, Cn and Cnv molecules (including the linear C∞v group) can be polar. In every other group the symmetry operations cancel any dipole. So water (C2v) and ammonia (C3v) are polar, while CO2 (D∞h), BF3 (D3h) and CH4 (Td) are not.

What does the order of a point group mean?

The order is the number of symmetry operations in the group, including the identity. C2v has order 4 (E, C2, σv, σv′), C3v has order 6, Td has 24 and Oh has 48. Linear groups have infinite order.

Why do staggered and eclipsed ethane have different point groups?

Point groups describe the 3D shape. Staggered ethane has a centre of inversion and dihedral planes, so it is D3d. Eclipsed ethane has a horizontal mirror plane instead, so it is D3h. When you enter a SMILES, the tool builds the lowest-energy shape, which for ethane is staggered.

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Last updated: October 18, 2026