In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and
inversion in a point on the axis. This is called an '''
n-fold improper rotation'
if the angle of rotation, before or after reflexion, is 360°/n
(where n'' must be even). There are several different systems for naming individual improper rotations: • In the
Schoenflies notation the symbol
Sn (German, ''
, for mirror), where n
must be even, denotes the symmetry group generated by an n
-fold improper rotation. For example, the symmetry operation S''6 is the combination of a rotation of (360°/6)=60° and a mirror plane reflection. (This should not be confused with the same notation for
symmetric groups). • In
Hermann–Mauguin notation the symbol is used for an '''
n-fold rotoinversion'
; i.e., rotation by an angle of rotation of 360°/n
with inversion. If n
is even it must be divisible by 4. (Note that would be simply a reflection, and is normally denoted "m", for "mirror".) When n
is odd this corresponds to a 2n''-fold improper rotation (or rotary reflexion). • The
Coxeter notation for
S2
n is [2
n+,2+] and , as an index 4 subgroup of [2
n,2], , generated as the product of 3 reflections. • The
Orbifold notation is
n×, order 2
n..
S2 is the
central inversion.
Cn are
cyclic groups.
Subgroups • The
direct subgroup of
S2
n is
Cn, order
n,
index 2, being the rotoreflection generator applied twice. • For odd
n,
S2
n contains an
inversion, denoted
Ci or
S2.
S2
n is the
direct product:
S2
n =
Cn ×
S2, if
n is odd. • For any
n, if odd
p is a divisor of
n, then
S2
n/
p is a subgroup of
S2
n, index
p. For example
S4 is a subgroup of
S12, index 3. ==As an indirect isometry ==