Dihedral symmetry in three dimensions

In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih"n" (for "n" ≥ 2).
Types.
There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, and orbifold notation.
For a given "n", all three have "n"-fold rotational symmetry about one axis (rotation by an angle of 360°/"n" does not change the object), and 2-fold rotational symmetry about a perpendicular axis, hence about "n" of those. For "n" = ∞, they correspond to three Frieze groups. Schönflies notation is used, with Coxeter notation in brackets, and orbifold notation in parentheses. The term horizontal (h) is used with respect to a vertical axis of rotation.
In 2D, the symmetry group "Dn" includes reflections in lines. When the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection through a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°. In 3D, the two operations are distinguished: the group "Dn" contains rotations only, not reflections. The other group is pyramidal symmetry "Cnv" of the same order, 2"n".
With reflection symmetry in a plane perpendicular to the "n"-fold rotation axis, we have "Dnh", [n], (*22"n").
"Dnd" (or "Dnv"), [2"n",2+], (2*"n") has vertical mirror planes between the horizontal rotation axes, not through them. As a result, the vertical axis is a 2"n"-fold rotoreflection axis.
"Dnh" is the symmetry group for a regular "n"-sided prism and also for a regular n-sided bipyramid. "Dnd" is the symmetry group for a regular "n"-sided antiprism, and also for a regular n-sided trapezohedron. "Dn" is the symmetry group of a partially rotated prism.
"n" = 1 is not included because the three symmetries are equal to other ones:
For "n" = 2 there is not one main axis and two additional axes, but there are three equivalent ones.
Subgroups.
For "Dnh", [n,2], (*22n), order 4n
For "Dnd", [2n,2+], (2*n), order 4n
"Dnd" is also subgroup of "D"2"nh".
Examples.
"Dnh", ["n"], (*22"n"):
"D"5"h", [5], (*225):
"D"4"d", [8,2+], (2*4):
"D"5"d", [10,2+], (2*5):
"D"17"d", [34,2+], (2*17):