<s>
It	O
is	O
a	O
subgroup	O
of	O
the	O
orthogonal	O
group	O
O(3 )	O
,	O
the	O
group	O
of	O
all	O
isometries	O
that	O
leave	O
the	O
origin	O
fixed	O
,	O
or	O
correspondingly	O
,	O
the	O
group	O
of	O
orthogonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
O(3 )	O
itself	O
is	O
a	O
subgroup	O
of	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
E(3 )	O
of	O
all	O
isometries	O
.	O
</s>
<s>
All	O
isometries	O
of	O
a	O
bounded	B-Algorithm
(	O
finite	O
)	O
3D	O
object	O
have	O
one	O
or	O
more	O
common	O
fixed	O
points	O
.	O
</s>
<s>
For	O
a	O
bounded	B-Algorithm
object	O
,	O
the	O
proper	O
symmetry	O
group	O
is	O
called	O
its	O
rotation	O
group	O
.	O
</s>
<s>
The	O
rotation	O
group	O
of	O
a	O
bounded	B-Algorithm
object	O
is	O
equal	O
to	O
its	O
full	O
symmetry	O
group	O
if	O
and	O
only	O
if	O
the	O
object	O
is	O
chiral	O
.	O
</s>
<s>
The	O
point	O
groups	O
that	O
are	O
generated	O
purely	O
by	O
a	O
finite	O
set	O
of	O
reflection	B-Algorithm
mirror	O
planes	O
passing	O
through	O
the	O
same	O
point	O
are	O
the	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
,	O
represented	O
by	O
Coxeter	O
notation	O
.	O
</s>
<s>
The	O
point	B-Algorithm
groups	I-Algorithm
in	I-Algorithm
three	I-Algorithm
dimensions	I-Algorithm
are	O
heavily	O
used	O
in	O
chemistry	O
,	O
especially	O
to	O
describe	O
the	O
symmetries	O
of	O
a	O
molecule	O
and	O
of	O
molecular	O
orbitals	O
forming	O
covalent	O
bonds	O
,	O
and	O
in	O
this	O
context	O
they	O
are	O
also	O
called	O
molecular	O
point	O
groups	O
.	O
</s>
<s>
The	O
identity	O
operation	O
,	O
denoted	O
by	O
E	O
or	O
the	O
identity	O
matrix	B-Architecture
I	O
.	O
</s>
<s>
Inversion	B-Algorithm
,	O
denoted	O
i	O
or	O
Ci	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
notation	O
is	O
−I	O
.	O
</s>
<s>
Reflection	B-Algorithm
in	O
a	O
plane	O
through	O
the	O
origin	O
,	O
denoted	O
σ	O
.	O
</s>
<s>
Improper	B-Algorithm
rotation	I-Algorithm
,	O
also	O
called	O
rotation-reflection	O
:	O
rotation	O
about	O
an	O
axis	O
by	O
an	O
angle	O
θ	O
,	O
combined	O
with	O
reflection	B-Algorithm
in	O
the	O
plane	O
through	O
the	O
origin	O
perpendicular	O
to	O
the	O
axis	O
.	O
</s>
<s>
Rotation-reflection	O
by	O
θ	O
=	O
360°	O
/n	O
for	O
any	O
positive	O
integer	O
n	O
is	O
denoted	O
Sn	O
(	O
from	O
the	O
Schoenflies	O
notation	O
for	O
the	O
group	O
Sn	O
that	O
it	O
generates	O
)	O
.	O
</s>
<s>
Inversion	B-Algorithm
is	O
a	O
special	O
case	O
of	O
rotation-reflection	O
(	O
i	O
=	O
S2	O
)	O
,	O
as	O
is	O
reflection	B-Algorithm
(σ=	O
S1	O
)	O
,	O
so	O
these	O
operations	O
are	O
often	O
considered	O
to	O
be	O
improper	B-Algorithm
rotations	I-Algorithm
.	O
</s>
<s>
The	O
other	O
infinite	O
isometry	O
groups	O
consist	O
of	O
all	O
rotations	O
about	O
an	O
axis	O
through	O
the	O
origin	O
,	O
and	O
those	O
with	O
additionally	O
reflection	B-Algorithm
in	O
the	O
planes	O
through	O
the	O
axis	O
,	O
and/or	O
reflection	B-Algorithm
in	O
the	O
plane	O
through	O
the	O
origin	O
,	O
perpendicular	O
to	O
the	O
axis	O
.	O
</s>
<s>
Those	O
with	O
reflection	B-Algorithm
in	O
the	O
planes	O
through	O
the	O
axis	O
,	O
with	O
or	O
without	O
reflection	B-Algorithm
in	O
the	O
plane	O
through	O
the	O
origin	O
perpendicular	O
to	O
the	O
axis	O
,	O
are	O
the	O
symmetry	O
groups	O
for	O
the	O
two	O
types	O
of	O
cylindrical	O
symmetry	O
.	O
</s>
<s>
Any	O
3D	O
shape	O
(	O
subset	O
of	O
R3	O
)	O
having	O
infinite	O
rotational	O
symmetry	O
must	O
also	O
have	O
mirror	B-Algorithm
symmetry	I-Algorithm
for	O
every	O
plane	O
through	O
the	O
axis	O
.	O
</s>
<s>
For	O
finite	O
3D	B-Algorithm
point	I-Algorithm
groups	I-Algorithm
,	O
see	O
also	O
spherical	O
symmetry	O
groups	O
.	O
</s>
<s>
Up	O
to	O
conjugacy	O
,	O
the	O
set	O
of	O
finite	O
3D	B-Algorithm
point	I-Algorithm
groups	I-Algorithm
consists	O
of	O
:	O
</s>
<s>
There	O
are	O
four	O
series	O
with	O
no	O
other	O
axes	O
of	O
rotational	O
symmetry	O
(	O
see	O
cyclic	O
symmetries	O
)	O
and	O
three	O
with	O
additional	O
axes	O
of	O
2-fold	O
symmetry	O
(	O
see	O
dihedral	B-Algorithm
symmetry	I-Algorithm
)	O
.	O
</s>
<s>
They	O
can	O
be	O
understood	O
as	O
point	B-Algorithm
groups	I-Algorithm
in	I-Algorithm
two	I-Algorithm
dimensions	I-Algorithm
extended	O
with	O
an	O
axial	O
coordinate	O
and	O
reflections	O
in	O
it	O
.	O
</s>
<s>
They	O
are	O
related	O
to	O
the	O
frieze	B-Algorithm
groups	I-Algorithm
;	O
they	O
can	O
be	O
interpreted	O
as	O
frieze-group	O
patterns	O
repeated	O
n	O
times	O
around	O
a	O
cylinder	O
.	O
</s>
<s>
The	O
orbifold	O
notation	O
is	O
a	O
unified	O
notation	O
,	O
also	O
applicable	O
for	O
wallpaper	O
groups	O
and	O
frieze	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Intl	O
Schoenflies	O
Orbifold	O
Coxeter	O
Frieze	B-Algorithm
Struct	O
.	O
</s>
<s>
Cs	O
(	O
equivalent	O
to	O
C1hand	O
C1v	O
)	O
–	O
reflection	B-Algorithm
symmetry	I-Algorithm
,	O
also	O
called	O
bilateral	O
symmetry	O
.	O
</s>
<s>
If	O
both	O
horizontal	O
and	O
vertical	O
reflection	B-Algorithm
planes	O
are	O
added	O
,	O
their	O
intersections	O
give	O
n	O
axes	O
of	O
rotation	O
through	O
180°	O
,	O
so	O
the	O
group	O
is	O
no	O
longer	O
uniaxial	O
.	O
</s>
<s>
Its	O
subgroup	O
of	O
rotations	O
is	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
Dn	O
of	O
order	O
2n	O
,	O
which	O
still	O
has	O
the	O
2-fold	O
rotation	O
axes	O
perpendicular	O
to	O
the	O
primary	O
rotation	O
axis	O
,	O
but	O
no	O
mirror	O
planes	O
.	O
</s>
<s>
There	O
is	O
one	O
more	O
group	O
in	O
this	O
family	O
,	O
called	O
Dnd	O
(	O
or	O
Dnv	O
)	O
,	O
which	O
has	O
vertical	O
mirror	O
planes	O
containing	O
the	O
main	O
rotation	O
axis	O
,	O
but	O
instead	O
of	O
having	O
a	O
horizontal	O
mirror	O
plane	O
,	O
it	O
has	O
an	O
isometry	O
that	O
combines	O
a	O
reflection	B-Algorithm
in	O
the	O
horizontal	O
plane	O
and	O
a	O
rotation	O
by	O
an	O
angle	O
180°	O
/n	O
.	O
</s>
<s>
The	O
group	O
Sn	O
is	O
generated	O
by	O
the	O
combination	O
of	O
a	O
reflection	B-Algorithm
in	O
the	O
horizontal	O
plane	O
and	O
a	O
rotation	O
by	O
an	O
angle	O
360°	O
/n	O
.	O
</s>
<s>
For	O
n	O
odd	O
this	O
is	O
equal	O
to	O
the	O
group	O
generated	O
by	O
the	O
two	O
separately	O
,	O
Cnh	O
of	O
order	O
2n	O
,	O
and	O
therefore	O
the	O
notation	O
Sn	O
is	O
not	O
needed	O
;	O
however	O
,	O
for	O
n	O
even	O
it	O
is	O
distinct	O
,	O
and	O
of	O
order	O
n	O
.	O
Like	O
Dnd	O
it	O
contains	O
a	O
number	O
of	O
improper	B-Algorithm
rotations	I-Algorithm
without	O
containing	O
the	O
corresponding	O
rotations	O
.	O
</s>
<s>
D1d	O
and	O
C2h	O
:	O
group	O
of	O
order	O
4	O
with	O
a	O
reflection	B-Algorithm
in	O
a	O
plane	O
and	O
a	O
180°	O
rotation	O
through	O
a	O
line	O
perpendicular	O
to	O
that	O
plane	O
.	O
</s>
<s>
S2	O
is	O
the	O
group	O
of	O
order	O
2	O
with	O
a	O
single	O
inversion	B-Algorithm
(	O
Ci	O
)	O
.	O
</s>
<s>
Generated	O
by	O
element	O
C2nσh	O
,	O
where	O
σh	O
is	O
a	O
reflection	B-Algorithm
in	O
the	O
direction	O
of	O
the	O
axis	O
.	O
</s>
<s>
Generated	O
by	O
element	O
Cn	O
and	O
reflection	B-Algorithm
σh	O
.	O
</s>
<s>
Generated	O
by	O
element	O
Cn	O
and	O
reflection	B-Algorithm
σv	O
in	O
a	O
direction	O
in	O
the	O
plane	O
perpendicular	O
to	O
the	O
axis	O
.	O
</s>
<s>
Here	O
,	O
Cn	O
denotes	O
an	O
axis	O
of	O
rotation	O
through	O
360°	O
/n	O
and	O
Sn	O
denotes	O
an	O
axis	O
of	O
improper	B-Algorithm
rotation	I-Algorithm
through	O
the	O
same	O
.	O
</s>
<s>
There	O
are	O
four	O
C3	O
axes	O
,	O
each	O
through	O
two	O
vertices	O
of	O
a	O
circumscribing	O
cube	B-Application
(	O
red	O
cube	B-Application
in	O
images	O
)	O
,	O
or	O
through	O
one	O
vertex	O
of	O
a	O
regular	O
tetrahedron	O
,	O
and	O
three	O
C2	O
axes	O
,	O
through	O
the	O
centers	O
of	O
the	O
cube	B-Application
's	O
faces	O
,	O
or	O
the	O
midpoints	O
of	O
the	O
tetrahedron	O
's	O
edges	O
.	O
</s>
<s>
This	O
group	O
is	O
isomorphic	O
to	O
A4	O
,	O
the	O
alternating	B-Algorithm
group	I-Algorithm
on	O
4	O
elements	O
,	O
and	O
is	O
the	O
rotation	O
group	O
for	O
a	O
regular	O
tetrahedron	O
.	O
</s>
<s>
This	O
group	O
has	O
six	O
mirror	O
planes	O
,	O
each	O
containing	O
two	O
edges	O
of	O
the	O
cube	B-Application
or	O
one	O
edge	O
of	O
the	O
tetrahedron	O
,	O
a	O
single	O
S4	O
axis	O
,	O
and	O
two	O
C3	O
axes	O
.	O
</s>
<s>
Td	O
is	O
isomorphic	O
to	O
S4	O
,	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
4	O
letters	O
,	O
because	O
there	O
is	O
a	O
1-to-1	O
correspondence	O
between	O
the	O
elements	O
of	O
Td	O
and	O
the	O
24	O
permutations	O
of	O
the	O
four	O
3-fold	O
axes	O
.	O
</s>
<s>
This	O
group	O
has	O
the	O
same	O
rotation	O
axes	O
as	O
T	O
,	O
with	O
mirror	O
planes	O
parallel	O
to	O
the	O
cube	B-Application
faces	O
.	O
</s>
<s>
The	O
C3	O
axes	O
become	O
S6	O
axes	O
,	O
and	O
there	O
is	O
inversion	B-Algorithm
symmetry	I-Algorithm
.	O
</s>
<s>
Th	O
is	O
isomorphic	O
to	O
A4	O
×	O
Z2	O
(	O
since	O
T	O
and	O
Ci	O
are	O
both	O
normal	O
subgroups	O
)	O
,	O
and	O
not	O
to	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
S4	O
.	O
</s>
<s>
It	O
is	O
the	O
symmetry	O
of	O
a	O
cube	B-Application
with	O
on	O
each	O
face	O
a	O
line	O
segment	O
dividing	O
the	O
face	O
into	O
two	O
equal	O
rectangles	O
,	O
such	O
that	O
the	O
line	O
segments	O
of	O
adjacent	O
faces	O
do	O
not	O
meet	O
at	O
the	O
edge	O
.	O
</s>
<s>
The	O
symmetries	O
correspond	O
to	O
the	O
even	O
permutations	O
of	O
the	O
body	O
diagonals	O
and	O
the	O
same	O
combined	O
with	O
inversion	B-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
the	O
symmetry	O
of	O
a	O
pyritohedron	O
,	O
which	O
is	O
similar	O
to	O
the	O
cube	B-Application
described	O
,	O
with	O
each	O
rectangle	O
replaced	O
by	O
a	O
pentagon	O
with	O
one	O
symmetry	O
axis	O
and	O
4	O
equal	O
sides	O
and	O
1	O
different	O
side	O
(	O
the	O
one	O
corresponding	O
to	O
the	O
line	O
segment	O
dividing	O
the	O
cube	B-Application
's	O
face	O
)	O
;	O
i.e.	O
,	O
the	O
cube	B-Application
's	O
faces	O
bulge	O
out	O
at	O
the	O
dividing	O
line	O
and	O
become	O
narrower	O
there	O
.	O
</s>
<s>
|align	O
=	O
center|O	O
,	O
(	O
432	O
)	O
[4,3]+	O
(	O
)	O
432order	O
24||chiral	O
octahedral	O
symmetry||This	O
group	O
is	O
like	O
T	O
,	O
but	O
the	O
C2	O
axes	O
are	O
now	O
C4	O
axes	O
,	O
and	O
additionally	O
there	O
are	O
6	O
C2	O
axes	O
,	O
through	O
the	O
midpoints	O
of	O
the	O
edges	O
of	O
the	O
cube	B-Application
.	O
</s>
<s>
It	O
is	O
the	O
rotation	O
group	O
of	O
the	O
cube	B-Application
and	O
octahedron	O
.	O
</s>
<s>
This	O
group	O
is	O
isomorphic	O
to	O
S4	O
×	O
Z2	O
(	O
because	O
both	O
O	O
and	O
Ci	O
are	O
normal	O
subgroups	O
)	O
,	O
and	O
is	O
the	O
symmetry	O
group	O
of	O
the	O
cube	B-Application
and	O
octahedron	O
.	O
</s>
<s>
See	O
also	O
the	O
isometries	O
of	O
the	O
cube	B-Application
.	O
</s>
<s>
The	O
group	O
I	O
is	O
isomorphic	O
to	O
A5	O
,	O
the	O
alternating	B-Algorithm
group	I-Algorithm
on	O
5	O
letters	O
,	O
since	O
its	O
elements	O
correspond	O
1-to-1	O
with	O
even	O
permutations	O
of	O
the	O
five	O
T	O
symmetries	O
(	O
or	O
the	O
five	O
tetrahedra	O
just	O
mentioned	O
)	O
.	O
</s>
<s>
The	O
reflective	O
point	B-Algorithm
groups	I-Algorithm
in	I-Algorithm
three	I-Algorithm
dimensions	I-Algorithm
are	O
also	O
called	O
Coxeter	B-Algorithm
groups	I-Algorithm
and	O
can	O
be	O
given	O
by	O
a	O
Coxeter-Dynkin	O
diagram	O
and	O
represent	O
a	O
set	O
of	O
mirrors	B-Application
that	O
intersect	O
at	O
one	O
central	O
point	O
.	O
</s>
<s>
A	O
rank	O
n	O
Coxeter	B-Algorithm
group	I-Algorithm
has	O
n	O
mirror	O
planes	O
.	O
</s>
<s>
Coxeter	B-Algorithm
groups	I-Algorithm
having	O
fewer	O
than	O
3	O
generators	O
have	O
degenerate	O
spherical	O
triangle	O
domains	O
,	O
as	O
lunes	O
or	O
a	O
hemisphere	O
.	O
</s>
<s>
In	O
Coxeter	O
notation	O
these	O
groups	O
are	O
tetrahedral	O
symmetry	O
[3,3],	O
octahedral	O
symmetry	O
[4,3],	O
icosahedral	O
symmetry	O
[5,3],	O
and	O
dihedral	B-Algorithm
symmetry	I-Algorithm
 [ p , 2 ] 	O
.	O
</s>
<s>
the	O
finite	O
subgroups	O
of	O
SO(3 )	O
,	O
are	O
:	O
the	O
cyclic	O
groups	O
Cn	O
(	O
the	O
rotation	O
group	O
of	O
a	O
canonical	O
pyramid	O
)	O
,	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
Dn	O
(	O
the	O
rotation	O
group	O
of	O
a	O
uniform	O
prism	O
,	O
or	O
canonical	O
bipyramid	O
)	O
,	O
and	O
the	O
rotation	O
groups	O
T	O
,	O
O	O
and	O
I	O
of	O
a	O
regular	O
tetrahedron	O
,	O
octahedron/cube	O
and	O
icosahedron/dodecahedron	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
D3	O
,	O
D4	O
etc	O
.	O
</s>
<s>
Therefore	O
,	O
it	O
is	O
also	O
called	O
a	O
dihedron	O
(	O
Greek	O
:	O
solid	O
with	O
two	O
faces	O
)	O
,	O
which	O
explains	O
the	O
name	O
dihedral	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Correspondingly	O
,	O
O(3 )	O
is	O
the	O
direct	O
product	O
of	O
SO(3 )	O
and	O
the	O
inversion	B-Algorithm
group	O
Ci	O
(	O
where	O
inversion	B-Algorithm
is	O
denoted	O
by	O
its	O
matrix	B-Architecture
−I	O
)	O
:	O
</s>
<s>
Thus	O
there	O
is	O
a	O
1-to-1	O
correspondence	O
between	O
all	O
direct	O
isometries	O
and	O
all	O
indirect	O
isometries	O
,	O
through	O
inversion	B-Algorithm
.	O
</s>
<s>
Also	O
there	O
is	O
a	O
1-to-1	O
correspondence	O
between	O
all	O
groups	O
H	O
of	O
direct	O
isometries	O
in	O
SO(3 )	O
and	O
all	O
groups	O
K	O
of	O
isometries	O
in	O
O(3 )	O
that	O
contain	O
inversion	B-Algorithm
:	O
</s>
<s>
If	O
a	O
group	O
of	O
direct	O
isometries	O
H	O
has	O
a	O
subgroup	O
L	O
of	O
index	O
2	O
,	O
then	O
there	O
is	O
a	O
corresponding	O
group	O
that	O
contains	O
indirect	O
isometries	O
but	O
no	O
inversion	B-Algorithm
:	O
</s>
<s>
This	O
group	O
M	O
is	O
,	O
when	O
considered	O
as	O
an	O
abstract	O
group	O
,	O
isomorphic	O
to	O
H	O
.	O
Conversely	O
,	O
for	O
all	O
point	O
groups	O
M	O
that	O
contain	O
indirect	O
isometries	O
but	O
no	O
inversion	B-Algorithm
we	O
can	O
obtain	O
a	O
rotation	O
group	O
H	O
by	O
inverting	O
the	O
indirect	O
isometries	O
.	O
</s>
<s>
Since	O
any	O
subgroup	O
of	O
index	O
two	O
is	O
normal	O
,	O
the	O
group	O
of	O
rotations	O
(	O
Cn	O
)	O
is	O
normal	O
both	O
in	O
the	O
group	O
(	O
Cnv	O
)	O
obtained	O
by	O
adding	O
to	O
(	O
Cn	O
)	O
reflection	B-Algorithm
planes	O
through	O
its	O
axis	O
and	O
in	O
the	O
group	O
(	O
Cnh	O
)	O
obtained	O
by	O
adding	O
to	O
(	O
Cn	O
)	O
a	O
reflection	B-Algorithm
plane	O
perpendicular	O
to	O
its	O
axis	O
.	O
</s>
<s>
For	O
even	O
order	O
2n	O
there	O
is	O
the	O
group	O
S2n	O
(	O
Schoenflies	O
notation	O
)	O
generated	O
by	O
a	O
rotation	O
by	O
an	O
angle	O
180°	O
/n	O
about	O
an	O
axis	O
,	O
combined	O
with	O
a	O
reflection	B-Algorithm
in	O
the	O
plane	O
perpendicular	O
to	O
the	O
axis	O
.	O
</s>
<s>
For	O
S2	O
the	O
notation	O
Ci	O
is	O
used	O
;	O
it	O
is	O
generated	O
by	O
inversion	B-Algorithm
.	O
</s>
<s>
For	O
any	O
order	O
2n	O
where	O
n	O
is	O
odd	O
,	O
we	O
have	O
Cnh	O
;	O
it	O
has	O
an	O
n-fold	O
rotation	O
axis	O
,	O
and	O
a	O
perpendicular	O
plane	O
of	O
reflection	B-Algorithm
.	O
</s>
<s>
It	O
is	O
generated	O
by	O
a	O
rotation	O
by	O
an	O
angle	O
360°	O
/n	O
about	O
the	O
axis	O
,	O
combined	O
with	O
the	O
reflection	B-Algorithm
.	O
</s>
<s>
For	O
C1h	O
the	O
notation	O
Cs	O
is	O
used	O
;	O
it	O
is	O
generated	O
by	O
reflection	B-Algorithm
in	O
a	O
plane	O
.	O
</s>
<s>
In	O
2D	O
dihedral	B-Algorithm
group	I-Algorithm
Dn	O
includes	O
reflections	O
,	O
which	O
can	O
also	O
be	O
viewed	O
as	O
flipping	O
over	O
flat	O
objects	O
without	O
distinction	O
of	O
front	O
-	O
and	O
backside	O
.	O
</s>
<s>
The	O
abstract	O
group	O
type	O
is	O
dihedral	B-Algorithm
group	I-Algorithm
Dihn	O
,	O
which	O
is	O
also	O
denoted	O
by	O
Dn	O
.	O
</s>
<s>
They	O
fit	O
anyway	O
if	O
the	O
fundamental	O
domain	O
is	O
bounded	B-Algorithm
by	O
reflection	B-Algorithm
planes	O
.	O
</s>
