If the figure has only one rotational axis, there are the four possible kinds of symmetries, all cyclic:

(there is an axis of rotation and reflection, but there is no mirror plane),

(there is a mirror plane, but it is not perpendicular to the axis),

(there is a mirror plane that is perpendicular to the axis), and

(there is a glide reflection).

If the figure has more than one rotational axis but no more than one

-fold axis with

, the possibilities are (dihedral symmetries):

(no mirror plane),

(the mirror plane is not perpendicular to the principal axis),

(the mirror plane is perpendicular to the principal axis).

The figure may have more than one 5-fold axis (icosahedral symmetry):

(rotations only),

(there is a mirror plane).

The figure may have more than one 4-fold axis (octahedral symmetry):

(rotations only),

(there is a mirror plane).

The figure may have more than one principal 3-fold axis (tetrahedral symmetry):

(rotations only),

(there is a mirror plane, no inversion),

(there is a point of inversion).

This Demonstration is a guessing game to learn about the 14 types of symmetry groups of figures that have a rotational axis.

Not demonstrated are the three symmetry groups with no rotational symmetry:

(asymmetric ),

(only inversion), and

(only one mirror plane).