The Bauhinia blakeana flower on the Hong Kong region flag has C_{5} symmetry; the star on each petal has D_{5} symmetry.  The Yin and Yang symbol has C_{2} symmetry of geometry with inverted colors 
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.
Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).
The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number of dimensions. These are the crystallographic point groups.
Point groups can be classified into chiral (or purely rotational) groups and achiral groups.^{ [1] } The chiral groups are subgroups of the special orthogonal group SO(d): they contain only orientationpreserving orthogonal transformations, i.e., those of determinant +1. The achiral groups contain also transformations of determinant −1. In an achiral group, the orientationpreserving transformations form a (chiral) subgroup of index 2.
Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a CoxeterDynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. Reflection groups are necessarily achiral (except for the trivial group containing only the identity element).
There are only two onedimensional point groups, the identity group and the reflection group.
Group  Coxeter  Coxeter diagram  Order  Description 

C_{1}  [ ]^{+}  1  Identity  
D_{1}  [ ]  2  Reflection group 
Point groups in two dimensions, sometimes called rosette groups.
They come in two infinite families:
Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups.
Group  Intl  Orbifold  Coxeter  Order  Description 

C_{n}  n  n•  [n]^{+}  n  Cyclic: nfold rotations. Abstract group Z_{n}, the group of integers under addition modulo n. 
D_{n}  nm  *n•  [n]  2n  Dihedral: cyclic with reflections. Abstract group Dih_{n}, the dihedral group. 
The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an isomorphism, mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order.
Reflective  Rotational  Related polygons  

Group  Coxeter group  Coxeter diagram  Order  Subgroup  Coxeter  Order  
D_{1}  A_{1}  [ ]  2  C_{1}  []^{+}  1  Digon  
D_{2}  A_{1}^{2}  [2]  4  C_{2}  [2]^{+}  2  Rectangle  
D_{3}  A_{2}  [3]  6  C_{3}  [3]^{+}  3  Equilateral triangle  
D_{4}  BC_{2}  [4]  8  C_{4}  [4]^{+}  4  Square  
D_{5}  H_{2}  [5]  10  C_{5}  [5]^{+}  5  Regular pentagon  
D_{6}  G_{2}  [6]  12  C_{6}  [6]^{+}  6  Regular hexagon  
D_{n}  I_{2}(n)  [n]  2n  C_{n}  [n]^{+}  n  Regular polygon  
D_{2}×2  A_{1}^{2}×2  [[2]] = [4]  =  8  
D_{3}×2  A_{2}×2  [[3]] = [6]  =  12  
D_{4}×2  BC_{2}×2  [[4]] = [8]  =  16  
D_{5}×2  H_{2}×2  [[5]] = [10]  =  20  
D_{6}×2  G_{2}×2  [[6]] = [12]  =  24  
D_{n}×2  I_{2}(n)×2  [[n]] = [2n]  =  4n 
Point groups in three dimensions, sometimes called molecular point groups after their wide use in studying symmetries of molecules.
They come in 7 infinite families of axial or prismatic groups, and 7 additional polyhedral or Platonic groups. In Schönflies notation,
Applying the crystallographic restriction theorem to these groups yields the 32 crystallographic point groups.

 
(*) When the Intl entries are duplicated, the first is for even n, the second for odd n. 
The reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The [3,3] group can be doubled, written as [[3,3]], mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group.
Schönflies  Coxeter group  Coxeter diagram  Order  Related regular and prismatic polyhedra  

T_{d}  A_{3}  [3,3]  24  Tetrahedron  
T_{d}×Dih_{1} = O_{h}  A_{3}×2 = BC_{3}  [[3,3]] = [4,3]  =  48  Stellated octahedron  
O_{h}  BC_{3}  [4,3]  48  Cube, octahedron  
I_{h}  H_{3}  [5,3]  120  Icosahedron, dodecahedron  
D_{3h}  A_{2}×A_{1}  [3,2]  12  Triangular prism  
D_{3h}×Dih_{1} = D_{6h}  A_{2}×A_{1}×2  [[3],2]  =  24  Hexagonal prism  
D_{4h}  BC_{2}×A_{1}  [4,2]  16  Square prism  
D_{4h}×Dih_{1} = D_{8h}  BC_{2}×A_{1}×2  [[4],2] = [8,2]  =  32  Octagonal prism  
D_{5h}  H_{2}×A_{1}  [5,2]  20  Pentagonal prism  
D_{6h}  G_{2}×A_{1}  [6,2]  24  Hexagonal prism  
D_{nh}  I_{2}(n)×A_{1}  [n,2]  4n  ngonal prism  
D_{nh}×Dih_{1} = D_{2nh}  I_{2}(n)×A_{1}×2  [[n],2]  =  8n  
D_{2h}  A_{1}^{3}  [2,2]  8  Cuboid  
D_{2h}×Dih_{1}  A_{1}^{3}×2  [[2],2] = [4,2]  =  16  
D_{2h}×Dih_{3} = O_{h}  A_{1}^{3}×6  [3[2,2]] = [4,3]  =  48  
C_{3v}  A_{2}  [1,3]  6  Hosohedron  
C_{4v}  BC_{2}  [1,4]  8  
C_{5v}  H_{2}  [1,5]  10  
C_{6v}  G_{2}  [1,6]  12  
C_{nv}  I_{2}(n)  [1,n]  2n  
C_{nv}×Dih_{1} = C_{2nv}  I_{2}(n)×2  [1,[n]] = [1,2n]  =  4n  
C_{2v}  A_{1}^{2}  [1,2]  4  
C_{2v}×Dih_{1}  A_{1}^{2}×2  [1,[2]]  =  8  
C_{s}  A_{1}  [1,1]  2 
The fourdimensional point groups (chiral as well as achiral) are listed in Conway and Smith,^{ [1] } Section 4, Tables 4.14.3.
The following list gives the fourdimensional reflection groups (excluding those that leave a subspace fixed and that are therefore lowerdimensional reflection groups). Each group is specified as a Coxeter group, and like the polyhedral groups of 3D, it can be named by its related convex regular 4polytope. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3]^{+} has three 3fold gyration points and symmetry order 60. Frontback symmetric groups like [3,3,3] and [3,4,3] can be doubled, shown as double brackets in Coxeter's notation, for example [[3,3,3]] with its order doubled to 240.
Coxeter group/notation  Coxeter diagram  Order  Related polytopes  

A_{4}  [3,3,3]  120  5cell  
A_{4}×2  [[3,3,3]]  240  5cell dual compound  
BC_{4}  [4,3,3]  384  16cell/Tesseract  
D_{4}  [3^{1,1,1}]  192  Demitesseractic  
D_{4}×2 = BC_{4}  <[3,3^{1,1}]> = [4,3,3]  =  384  
D_{4}×6 = F_{4}  [3[3^{1,1,1}]] = [3,4,3]  =  1152  
F_{4}  [3,4,3]  1152  24cell  
F_{4}×2  [[3,4,3]]  2304  24cell dual compound  
H_{4}  [5,3,3]  14400  120cell/600cell  
A_{3}×A_{1}  [3,3,2]  48  Tetrahedral prism  
A_{3}×A_{1}×2  [[3,3],2] = [4,3,2]  =  96  Octahedral prism  
BC_{3}×A_{1}  [4,3,2]  96  
H_{3}×A_{1}  [5,3,2]  240  Icosahedral prism  
A_{2}×A_{2}  [3,2,3]  36  Duoprism  
A_{2}×BC_{2}  [3,2,4]  48  
A_{2}×H_{2}  [3,2,5]  60  
A_{2}×G_{2}  [3,2,6]  72  
BC_{2}×BC_{2}  [4,2,4]  64  
BC_{2}^{2}×2  [[4,2,4]]  128  
BC_{2}×H_{2}  [4,2,5]  80  
BC_{2}×G_{2}  [4,2,6]  96  
H_{2}×H_{2}  [5,2,5]  100  
H_{2}×G_{2}  [5,2,6]  120  
G_{2}×G_{2}  [6,2,6]  144  
I_{2}(p)×I_{2}(q)  [p,2,q]  4pq  
I_{2}(2p)×I_{2}(q)  [[p],2,q] = [2p,2,q]  =  8pq  
I_{2}(2p)×I_{2}(2q)  [[p]],2,[[q]] = [2p,2,2q]  =  16pq  
I_{2}(p)^{2}×2  [[p,2,p]]  8p^{2}  
I_{2}(2p)^{2}×2  [[[p],2,[p]]] = [[2p,2,2p]]  =  32p^{2}  
A_{2}×A_{1}×A_{1}  [3,2,2]  24  
BC_{2}×A_{1}×A_{1}  [4,2,2]  32  
H_{2}×A_{1}×A_{1}  [5,2,2]  40  
G_{2}×A_{1}×A_{1}  [6,2,2]  48  
I_{2}(p)×A_{1}×A_{1}  [p,2,2]  8p  
I_{2}(2p)×A_{1}×A_{1}×2  [[p],2,2] = [2p,2,2]  =  16p  
I_{2}(p)×A_{1}^{2}×2  [p,2,[2]] = [p,2,4]  =  16p  
I_{2}(2p)×A_{1}^{2}×4  [[p]],2,[[2]] = [2p,2,4]  =  32p  
A_{1}×A_{1}×A_{1}×A_{1}  [2,2,2]  16  4orthotope  
A_{1}^{2}×A_{1}×A_{1}×2  [[2],2,2] = [4,2,2]  =  32  
A_{1}^{2}×A_{1}^{2}×4  [[2]],2,[[2]] = [4,2,4]  =  64  
A_{1}^{3}×A_{1}×6  [3[2,2],2] = [4,3,2]  =  96  
A_{1}^{4}×24  [3,3[2,2,2]] = [4,3,3]  =  384 
The following table gives the fivedimensional reflection groups (excluding those that are lowerdimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3]^{+} has four 3fold gyration points and symmetry order 360.
Coxeter group/notation  Coxeter diagrams  Order  Related regular and prismatic polytopes  

A_{5}  [3,3,3,3]  720  5simplex  
A_{5}×2  [[3,3,3,3]]  1440  5simplex dual compound  
BC_{5}  [4,3,3,3]  3840  5cube, 5orthoplex  
D_{5}  [3^{2,1,1}]  1920  5demicube  
D_{5}×2  <[3,3,3^{1,1}]>  =  3840  
A_{4}×A_{1}  [3,3,3,2]  240  5cell prism  
A_{4}×A_{1}×2  [[3,3,3],2]  480  
BC_{4}×A_{1}  [4,3,3,2]  768  tesseract prism  
F_{4}×A_{1}  [3,4,3,2]  2304  24cell prism  
F_{4}×A_{1}×2  [[3,4,3],2]  4608  
H_{4}×A_{1}  [5,3,3,2]  28800  600cell or 120cell prism  
D_{4}×A_{1}  [3^{1,1,1},2]  384  Demitesseract prism  
A_{3}×A_{2}  [3,3,2,3]  144  Duoprism  
A_{3}×A_{2}×2  [[3,3],2,3]  288  
A_{3}×BC_{2}  [3,3,2,4]  192  
A_{3}×H_{2}  [3,3,2,5]  240  
A_{3}×G_{2}  [3,3,2,6]  288  
A_{3}×I_{2}(p)  [3,3,2,p]  48p  
BC_{3}×A_{2}  [4,3,2,3]  288  
BC_{3}×BC_{2}  [4,3,2,4]  384  
BC_{3}×H_{2}  [4,3,2,5]  480  
BC_{3}×G_{2}  [4,3,2,6]  576  
BC_{3}×I_{2}(p)  [4,3,2,p]  96p  
H_{3}×A_{2}  [5,3,2,3]  720  
H_{3}×BC_{2}  [5,3,2,4]  960  
H_{3}×H_{2}  [5,3,2,5]  1200  
H_{3}×G_{2}  [5,3,2,6]  1440  
H_{3}×I_{2}(p)  [5,3,2,p]  240p  
A_{3}×A_{1}^{2}  [3,3,2,2]  96  
BC_{3}×A_{1}^{2}  [4,3,2,2]  192  
H_{3}×A_{1}^{2}  [5,3,2,2]  480  
A_{2}^{2}×A_{1}  [3,2,3,2]  72  duoprism prism  
A_{2}×BC_{2}×A_{1}  [3,2,4,2]  96  
A_{2}×H_{2}×A_{1}  [3,2,5,2]  120  
A_{2}×G_{2}×A_{1}  [3,2,6,2]  144  
BC_{2}^{2}×A_{1}  [4,2,4,2]  128  
BC_{2}×H_{2}×A_{1}  [4,2,5,2]  160  
BC_{2}×G_{2}×A_{1}  [4,2,6,2]  192  
H_{2}^{2}×A_{1}  [5,2,5,2]  200  
H_{2}×G_{2}×A_{1}  [5,2,6,2]  240  
G_{2}^{2}×A_{1}  [6,2,6,2]  288  
I_{2}(p)×I_{2}(q)×A_{1}  [p,2,q,2]  8pq  
A_{2}×A_{1}^{3}  [3,2,2,2]  48  
BC_{2}×A_{1}^{3}  [4,2,2,2]  64  
H_{2}×A_{1}^{3}  [5,2,2,2]  80  
G_{2}×A_{1}^{3}  [6,2,2,2]  96  
I_{2}(p)×A_{1}^{3}  [p,2,2,2]  16p  
A_{1}^{5}  [2,2,2,2]  32  5orthotope  
A_{1}^{5}×(2!)  [[2],2,2,2]  =  64  
A_{1}^{5}×(2!×2!)  [[2]],2,[2],2]  =  128  
A_{1}^{5}×(3!)  [3[2,2],2,2]  =  192  
A_{1}^{5}×(3!×2!)  [3[2,2],2,[[2]]  =  384  
A_{1}^{5}×(4!)  [3,3[2,2,2],2]]  =  768  
A_{1}^{5}×(5!)  [3,3,3[2,2,2,2]]  =  3840 
The following table gives the sixdimensional reflection groups (excluding those that are lowerdimensional reflection groups), by listing them as Coxeter groups. Related pure rotational groups exist for each with half the order, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3]^{+} has five 3fold gyration points and symmetry order 2520.
Coxeter group  Coxeter diagram  Order  Related regular and prismatic polytopes  

A_{6}  [3,3,3,3,3]  5040 (7!)  6simplex  
A_{6}×2  [[3,3,3,3,3]]  10080 (2×7!)  6simplex dual compound  
BC_{6}  [4,3,3,3,3]  46080 (2^{6}×6!)  6cube, 6orthoplex  
D_{6}  [3,3,3,3^{1,1}]  23040 (2^{5}×6!)  6demicube  
E_{6}  [3,3^{2,2}]  51840 (72×6!)  1_{22}, 2_{21}  
A_{5}×A_{1}  [3,3,3,3,2]  1440 (2×6!)  5simplex prism  
BC_{5}×A_{1}  [4,3,3,3,2]  7680 (2^{6}×5!)  5cube prism  
D_{5}×A_{1}  [3,3,3^{1,1},2]  3840 (2^{5}×5!)  5demicube prism  
A_{4}×I_{2}(p)  [3,3,3,2,p]  240p  Duoprism  
BC_{4}×I_{2}(p)  [4,3,3,2,p]  768p  
F_{4}×I_{2}(p)  [3,4,3,2,p]  2304p  
H_{4}×I_{2}(p)  [5,3,3,2,p]  28800p  
D_{4}×I_{2}(p)  [3,3^{1,1},2,p]  384p  
A_{4}×A_{1}^{2}  [3,3,3,2,2]  480  
BC_{4}×A_{1}^{2}  [4,3,3,2,2]  1536  
F_{4}×A_{1}^{2}  [3,4,3,2,2]  4608  
H_{4}×A_{1}^{2}  [5,3,3,2,2]  57600  
D_{4}×A_{1}^{2}  [3,3^{1,1},2,2]  768  
A_{3}^{2}  [3,3,2,3,3]  576  
A_{3}×BC_{3}  [3,3,2,4,3]  1152  
A_{3}×H_{3}  [3,3,2,5,3]  2880  
BC_{3}^{2}  [4,3,2,4,3]  2304  
BC_{3}×H_{3}  [4,3,2,5,3]  5760  
H_{3}^{2}  [5,3,2,5,3]  14400  
A_{3}×I_{2}(p)×A_{1}  [3,3,2,p,2]  96p  Duoprism prism  
BC_{3}×I_{2}(p)×A_{1}  [4,3,2,p,2]  192p  
H_{3}×I_{2}(p)×A_{1}  [5,3,2,p,2]  480p  
A_{3}×A_{1}^{3}  [3,3,2,2,2]  192  
BC_{3}×A_{1}^{3}  [4,3,2,2,2]  384  
H_{3}×A_{1}^{3}  [5,3,2,2,2]  960  
I_{2}(p)×I_{2}(q)×I_{2}(r)  [p,2,q,2,r]  8pqr  Triaprism  
I_{2}(p)×I_{2}(q)×A_{1}^{2}  [p,2,q,2,2]  16pq  
I_{2}(p)×A_{1}^{4}  [p,2,2,2,2]  32p  
A_{1}^{6}  [2,2,2,2,2]  64  6orthotope 
The following table gives the sevendimensional reflection groups (excluding those that are lowerdimensional reflection groups), by listing them as Coxeter groups. Related chiral groups exist for each with half the order, defined by an even number of reflections, and can be represented by the bracket Coxeter notation with a '+' exponent, for example [3,3,3,3,3,3]^{+} has six 3fold gyration points and symmetry order 20160.
Coxeter group  Coxeter diagram  Order  Related polytopes  

A_{7}  [3,3,3,3,3,3]  40320 (8!)  7simplex  
A_{7}×2  [[3,3,3,3,3,3]]  80640 (2×8!)  7simplex dual compound  
BC_{7}  [4,3,3,3,3,3]  645120 (2^{7}×7!)  7cube, 7orthoplex  
D_{7}  [3,3,3,3,3^{1,1}]  322560 (2^{6}×7!)  7demicube  
E_{7}  [3,3,3,3^{2,1}]  2903040 (8×9!)  3_{21}, 2_{31}, 1_{32}  
A_{6}×A_{1}  [3,3,3,3,3,2]  10080 (2×7!)  
BC_{6}×A_{1}  [4,3,3,3,3,2]  92160 (2^{7}×6!)  
D_{6}×A_{1}  [3,3,3,3^{1,1},2]  46080 (2^{6}×6!)  
E_{6}×A_{1}  [3,3,3^{2,1},2]  103680 (144×6!)  
A_{5}×I_{2}(p)  [3,3,3,3,2,p]  1440p  
BC_{5}×I_{2}(p)  [4,3,3,3,2,p]  7680p  
D_{5}×I_{2}(p)  [3,3,3^{1,1},2,p]  3840p  
A_{5}×A_{1}^{2}  [3,3,3,3,2,2]  2880  
BC_{5}×A_{1}^{2}  [4,3,3,3,2,2]  15360  
D_{5}×A_{1}^{2}  [3,3,3^{1,1},2,2]  7680  
A_{4}×A_{3}  [3,3,3,2,3,3]  2880  
A_{4}×BC_{3}  [3,3,3,2,4,3]  5760  
A_{4}×H_{3}  [3,3,3,2,5,3]  14400  
BC_{4}×A_{3}  [4,3,3,2,3,3]  9216  
BC_{4}×BC_{3}  [4,3,3,2,4,3]  18432  
BC_{4}×H_{3}  [4,3,3,2,5,3]  46080  
H_{4}×A_{3}  [5,3,3,2,3,3]  345600  
H_{4}×BC_{3}  [5,3,3,2,4,3]  691200  
H_{4}×H_{3}  [5,3,3,2,5,3]  1728000  
F_{4}×A_{3}  [3,4,3,2,3,3]  27648  
F_{4}×BC_{3}  [3,4,3,2,4,3]  55296  
F_{4}×H_{3}  [3,4,3,2,5,3]  138240  
D_{4}×A_{3}  [3^{1,1,1},2,3,3]  4608  
D_{4}×BC_{3}  [3,3^{1,1},2,4,3]  9216  
D_{4}×H_{3}  [3,3^{1,1},2,5,3]  23040  
A_{4}×I_{2}(p)×A_{1}  [3,3,3,2,p,2]  480p  
BC_{4}×I_{2}(p)×A_{1}  [4,3,3,2,p,2]  1536p  
D_{4}×I_{2}(p)×A_{1}  [3,3^{1,1},2,p,2]  768p  
F_{4}×I_{2}(p)×A_{1}  [3,4,3,2,p,2]  4608p  
H_{4}×I_{2}(p)×A_{1}  [5,3,3,2,p,2]  57600p  
A_{4}×A_{1}^{3}  [3,3,3,2,2,2]  960  
BC_{4}×A_{1}^{3}  [4,3,3,2,2,2]  3072  
F_{4}×A_{1}^{3}  [3,4,3,2,2,2]  9216  
H_{4}×A_{1}^{3}  [5,3,3,2,2,2]  115200  
D_{4}×A_{1}^{3}  [3,3^{1,1},2,2,2]  1536  
A_{3}^{2}×A_{1}  [3,3,2,3,3,2]  1152  
A_{3}×BC_{3}×A_{1}  [3,3,2,4,3,2] 