Basic group theory, symmetry
“The beauty and strength of group theory applied to physics resides in the transformation
of many complex symmetry operations into a fairly simple linear algebra.”
• Definition of a group
• Example of a group: P(3)
• P(3) as the symmetry group of equilateral triangle C3v
• Further terminology and properties, useful!
• Representations (efficient unified handling)
• Characters (distillation of the most important properties)
• Point groups (symmetry of molecules and crystals)
Fyzika kond. látek 2, jarní semestr 2021/2022 J. Humlíček, humlicek@physics.muni.cz
Group
A set G of elements A, B, … with the binary operation (AB, called multiplication)
having the following properties:
1. AB G (closure),
2. (AB)C=A(BC) (associativity),
3. G contains E such that EA=AE=A for any A G (E is unit element),
4. for every A G there exists A-1 such that AA-1= A-1A=E (A-1 is inverse of A).
Notes:
The sequence of listing the elements does not matter.
The multiplication need not be commutative (AB=BA).
If it is, the group is called Abelian.
Example of a group: permutations of three symbols P(3)
3!=6 elements (the group is of order 6),
E=(123) A=(132) B=(321) C=(213) D=(312) F=(231)
mean the final arrangement of the three symbols from the initial (123)
the multiplication means the sequential permutation of the three symbols: AD ... first D, then A
Multiplication table
right
left
E A B C D F
E E A B C D F
A A E D F B C
B B F E D C A
C C D F E A B
D D C A B F E
F F B C A E D
P(3) as (3x3) matrix multiplications (“row times column”)
E=(123) A=(132) B=(321)
1 1 0 0 1
2 0 1 0 2
3 0 0 1 3
     
          
          
1 1 0 0 1
3 0 0 1 2
2 0 1 0 3
     
          
          
3 0 0 1 1
2 0 1 0 2
1 1 0 0 3
     
          
          
C=(213) D=(312) F=(231)
2 0 1 0 1
1 1 0 0 2
3 0 0 1 3
     
          
          
3 0 0 1 1
1 1 0 0 2
2 0 1 0 3
     
          
          
2 0 1 0 1
3 0 0 1 2
1 1 0 0 3
     
          
          
1 0 0 0 0 1 0 0 1
0 0 1 1 0 0 0 1 0
0 1 0 0 1 0 1 0 0
AD B
     
            
          
0 0 1 1 0 0 0 1 0
1 0 0 0 0 1 1 0 0
0 1 0 0 1 0 0 0 1
DA C
     
            
          
P(3) as the set of (2x2) matrices (can these permute three objects?)
1 0
0 1
E
 
  
 
-1 0
0 1
A
 
  
 
1 - 31
2 - 3 -1
B
 
  
  
1 31
2 3 -1
C
 
  
  
-1 31
2 3 -1
D
 
  
  
-1 - 31
2 3 -1
F
 
  
  
-1 0 -1 3 1 - 31 1
0 1 2 2- 3 -1 - 3 -1
AD B
    
      
        
P(3) represents the symmetry operations on an equilateral triangle, the group C3v
1 0
0 1
E
 
  
 
-1 0
0 1
A
 
  
 
1 - 31
2 - 3 -1
B
 
  
  
1 31
2 3 -1
C
 
  
  
-1 31
2 3 -1
D
 
  
  
-1 - 31
2 3 -1
F
 
  
  
3
y
x
1
2
no change
23, mirror reflection in (x,z),(1,z)
13, mirror reflection in (2,z)
12, mirror reflection in (3,z)
1,2,33,1,2,
rotation by 2/3 about z
1,2,32,3,1,
rotation by 4/3 about z
-1 0
0 1
A
A
y y y
x x x
       
        
      
Further terminology and properties
1. The order of G: the number of its elements; P(3) is of order 6.
2. A subgroup of G: a set of its elements forming a group;
the subgroups of P(3): (E), (E,A), (E,B), (E,C), (E,D,F).
The order of a subgroup is a divisor of the order of the group.
3. The order n of AG: the smallest value in An=E;
in P(3), E is of order 1, A,B,C are of order 2, D,F are of order 3.
4. The period of AG is the Abelian subgroup (E,A, A2 ,..., An-1), where n is the
order of A; the periods of P(3): (E), (E,A), (E,B), (E,C), (E,D,D2)=(E, F,F2)
=(E,D,F).
Further terminology and properties, continued
5. Let S(E,S1,S2,...,Sg) be a subgroup of G and XG. The right coset of S is the set
(EX, S1X,S2X, ...,SgX), the left coset is (XE,XS1,XS2, ...,XSg). A coset need not be
a group. A coset will itself be a subgroup S if XS. Two cosets of a given
subgroup either contain the same elements, or have no elements in common.
Examples using P(3): Let S(E,A). The right cosets of S are (E,A)E=
(E,A)A=(E,A) which is a subgroup, and (E,A)B= (E,A)D= (B,D) and (E,A)C=
(E,A)F= (C,F) which are not subgroups. The left cosets of (E,A) are (E,A), (C,D)
and (B,F).
6. An element BG is called conjugate to AG if B=XAX-1, where X is an arbitrary
element of G. If B is conjugate to A and C is conjugate to B , then C is conjugate
to A.
7. A class is the set of elements of which is obtained from a given element of G by
conjugation. The unit element is the only class forming a subgroup. All elements
of a class have the same order. An Abelian group has as many classes as
elements. There are three classes in P(3): (E), (A,B,C), and (D,F).
The notion of class is very important; checking the classes of P(3):
class (a helpful look at the inverses, A-1=A, B-1=B, C-1=C, D-1=F, F-1=D):
(E) trivial, since XEX-1=E for any X G,
(A,B,C) elements of the order of 2, red in the multiplication table,
(D,F) elements of the order of 3, green in the multiplication table;
if one product in the conjugation is outside a class, the other “brings it back”
right
left
E A B C D F
E E A B C D F
A A E D F B C
B B F E D C A
C C D F E A B
D D C A B F E
F F B C A E D
Further terminology and properties, continued
8. A subgroup N of G is called self—conjugate (or invariant, or normal)
if XNX-1=N, where X is an arbitrary element of G.
To obtain a self—conjugate subgroup, it is necessary to include entire classes in
it. The right and left cosets of a self—conjugate subgroup are the same. The
products of the elements of two right cosets of a self—conjugate subgroup form
another right coset.
For example, (E,D,F) is a self—conjugate subgroup of P(3), while (E,A), (E,B),
(E,C) are not; the right and left cosets (E,D,F)A=(A,C,B) and A(E,D,F)=(A,B,C)
are the same; the products (A,C,B)(E,D,F) form the right coset (A,B,C), the
products (A,B,C)(A,B,C) form the right coset (E,D,F).
9. A group with no self—conjugate subgroups is called simple group.
Further terminology and properties, continued
10. The factor group of G results from a self—conjugate subgroup N as the set of its
cosets, i.e., each coset being considered an element of the factor group.
N is sometimes called a normal divisor.
Obviously, all four rules of the multiplication are satisfied:
1. The closure: (NiX)(NjY)=Ni(XNj )Y=Ni(NkX)Y=(NiNk)(XY) for X,YG and
Ni,Nj,NkN, Nk=XNj X-1.
2. The associativity holds because it holds for the elements of G.
3. The unit element of the factor group is the coset containing EG.
4. The inverse element exists: (XN)(X-1N)= (NX)(X-1N)=NN=N.
11. The index of a subgroup is the total number of cosets.
The order of the factor group is the index of the self—conjugate subgroup.
Further terminology and properties, continued
For example, E=(E,D,F) is a self—conjugate subgroup of P(3),
A=(A,B,C) is the only other (both right and left) coset.
E and A form the (Abelian) factor group with the multiplication table
E A
E E A
A A E
This group is isomorphic (equivalent, one—to—one correspondence) with the
group P(2) of the permutations of 2 objects, or with the subgroups (E,A), (E,B), (E,C)
of P(3).
Representation theory
Two groups, G=(A,B,C,D,...) and g=(u,v,...) are homomorphic, if there exists
a mapping (correspondence) of G into g, e.g.,
Au, (B,C) v,...,
such that
ABuv, AC uv,...
If the correspondence is one—to—one (the orders of G and g are the same), the two
groups are isomorphic.
For example, the mapping of P(3) into P(2),
(E,D,F) E, (A,B,C) A
is homomorphism. The correspondence of the permutation group P(3) with the
symmetry group of an equilateral triangle is isomorphism.
Representation theory
Homomorphism or isomorphism of a group G with a group of square matrices is
called a representation of G.
A matrix D(A) is assigned to each AG such that D(AB)=D(A)D(B), with the usual
way of multiplication of square matrices (“row times column”).
An example of a homomorphic representation of the permutation group P(3) is the
group of two one—dimensional matrices [1], [-1]:
(E,D,F) [1], (A,B,C) [-1] .
The multiplication table of this matrix group is
[1] [-1]
[1] [1] [-1]
[-1] [-1] [1]
Representation theory
Another easily found homomorphic representation of the permutation group P(3) is
the group consisting of the single one—dimensional matrix [1]:
(E,A,B,C, D,F) [1].
The multiplication table of this matrix group is
[1]
[1] [1]
The one—dimensional representation [1] is a representation of any group.
1 0 0
( ) 0 1 0
0 0 1
D E
 
   
  
1 0 0
( ) 0 0 1
0 1 0
D A
 
   
  
0 0 1
( ) 0 1 0
1 0 0
D B
 
   
  
0 1 0
( ) 1 0 0
0 0 1
D C
 
   
  
0 0 1
( ) 1 0 0
0 1 0
D D
 
   
  
0 1 0
( ) 0 0 1
1 0 0
D F
 
   
  
Representation theory
An isomorphic representation of the permutation group P(3) is the group of
the following six three—dimensional matrices:
(they permute the elements of column or row vectors by the usual multiplication)
Representation theory
Another isomorphic representation of the permutation group P(3) is the group of
the following six two—dimensional matrices:
(remember the transformation of the positions of vertices of the equilateral triangle)
1 0
( )
0 1
D E
 
  
 
-1 0
( )
0 1
D A
 
  
 
1 - 31
( )
2 - 3 -1
D B
 
  
  
1 31
( )
2 3 -1
D C
 
  
  
-1 31
( )
2 3 -1
D D
 
  
  
-1 - 31
( )
2 3 -1
D F
 
  
  
Representation theory
In general, the representations are built from square matrices with complex elements.
The symbol * denotes complex conjugation, T denotes transposition, and + denotes
the adjoint:
11 12
21 22
...
... ,
...
d d
D d d
 
   
  
11 21
12 22
...
... ,
...
T
d d
D d d
 
   
  
* *
11 21
* *
12 22
...
... .
...
d d
D d d
 
 
  
 
 
Hermitian matrices are defined by DT=D*, or D+=D.
Unitary matrices are defined by D+=D-1.
Unitary matrices preserve the norms of vectors:
(Dv)+Dv = (v+D+)(Dv) = v+v, where v is a column vector, v+ is the adjoint row
vector.
The dimensionality of a representation is the dimensionality (the number of rows
and columns) of its matrices.
Representation theory
The representations are not unique. A simple way of generating a new set of
representing matrices is combining them according to
(A)
( ) ,
(A)
n
T
m
D O
D A
O D
 
  
  
where Dn(A) and Dm (A) are matrices of an n— and m—dimensional representations,
respectively, and O is a nm matrix of zeros. The block form of the above D(A)
matrices means, that this representation is reducible (it contains at least two
representations). The zero matrix “separates” the two representations in the combined
matrices.
Further, performing a similarity transformation
UD(A)U-1
provides a new representation, called equivalent.
Representation theory
If a similarity transformation leads to the same block form of the matrices, the
representation is called reducible; otherwise it is irreducible. In other words, an
irreducible representation cannot be expressed in terms of lower—dimensional
representations.
Three irreducible representations of P(3) are:
group element:
symbol (label)
E A B C D F
G1 [1] [1] [1] [1] [1] [1]
G1’ [1] [-1] [-1] [-1] [1] [1]
G2
-1 0
0 1
 
 
 
1 0
0 1
 
 
 
1 - 31
2 - 3 -1
 
 
  
1 31
2 3 -1
 
 
  
-1 31
2 3 -1
 
 
  
-1 - 31
2 3 -1
 
 
  
Representation theory
An example of a reducible representation GR of P(3) is:
E A B ...
GR
...
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
 
 
 
 
 
 
1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 1
 
 
 
 
 
 
1 0 0 0
0 -1 0 0
1 - 3
0 0
2 2
- 3 1
0 0 -
2 2
 
 
 
 
 
 
 
 
  
The irreducible representations contained in the reducible representation GR
are usually listed as:
GR = G1 + G1’ + G2 .
Representation theory
Among different representations, those formed by unitary matrices are of
special importance. Every representation (D(Aj), j=1,...,n) with nonvanishing
determinants can be brought into the unitary form by a similarity transformation.
This is easily seen by inspecting the Hermitian matrix
which can be diagonalized by the unitary transformation U formed from the set
of orthonormal eigenvectors of H:
1
,
n
j j
j
H D D

 
1
.dH U HU

All elements of the diagonal matrix Hd are positive; we can therefore construct
its square root and find the wanted similarity transformation as:
1/2 1 1/2
( ) ( ) .u j d j dD A H U D A UH 

Representation theory
It can easily be shown that the matrices Du are unitary:
1/2 1 1/2 1/2 1 1/2
1/2 1 1 1/2
1/2 1 1 1 1 1/2
1
1/2 1 1 1/2
1
( ) ( ) ( ) ( ( ) )
( ) ( )
( ) [ ( ) ( ) ] ( )
[ ( ) ( ) ( ) ( ) ]
u j u j d j d d j d
d j d j d
n
d j k k j d
k
n
d j k k j d
k
D A D A H U D A UH H U D A UH
H U D A UH U D A UH
H U D A U U D A UU D A U U D A UH
H U D A D A UU D A D A U H
     
    
       

     




 

 ,I
where I is the unit matrix. The sum in the last row is obviously the diagonal
matrix Hd, since the multiplication by the fixed matrix D(Aj) merely rearranges
the elements of the representation.
Representation theory
The irreducible representation has the following two properties concerning the
commutation properties,
( ) ( ) , 1,..., ,j jMD A D A M j n 
with auxiliary matrices M (Schur’s first and second lemma):
1. The only matrix commuting with all matrices of the irreducible representation
is a const.I. If a non—constant commuting matrix exists, the representation is
reducible.
2. If the representations (Da(Aj), j=1,...,n) and (Db(Aj), j=1,...,n) of a given group
are of dimensionalities da and db, respectively, then if there is a dadb matrix
M
such that ( ) ( ) , 1,..., ,a j b jMD A D A M j n 
then M must be the null matrix if a da≠db . For a da=db , M is nonzero only for the
two representations differing by a similarity transformation, i.e., equivalent.
Representation theory - orthogonality of irreducible representations
The irreducible representations (Da(Aj), j=1,...,n) and (Db(Aj), j=1,...,n) of a given
group, of dimensionalities da and db, respectively, obey the following
great orthogonality theorem:
For unitary representations, the above relation simplifies to
1
, ,
1
( ) ( ) ,
n
a rs j b tu j ab rt su
j a
n
D A D A
d
  


where Da,rs is the element of Da from the r-th row and s-th column, and
ab the Kronecker symbol: ab =1 for a=b and ab =0 for a≠b.
*
, ,
1
( ) ( ) .
n
a rs j b tu j ab rt su
j a
n
D A D A
d
  

 (GOT)
Representation theory- orthogonality of irreducible representations
The irreducible representations of P(3):
E A B C D F
G1 [1] [1] [1] [1] [1] [1]
G1’ [1] [-1] [-1] [-1] [1] [1]
G2
-1 0
0 1
 
 
 
1 0
0 1
 
 
 
1 - 31
2 - 3 -1
 
 
  
1 31
2 3 -1
 
 
  
-1 31
2 3 -1
 
 
  
-1 - 31
2 3 -1
 
 
  
They are orthogonal:
1. The sum in (GOT) for G1 and G1’ vanishes: as many [1]’s as [-1]’s, the sum is [0].
2. The sum in (GOT) for G1 and G2 vanishes:
1 0 -1 0 1 - 3 1 3 -1 3 -1 - 3 0 01 1 1 1
.
0 1 0 1 0 02 2 2 2- 3 -1 3 -1 - 3 -1 3 -1
            
                 
                   
Representation theory - orthogonality of irreducible representations
The irreducible representations of P(3):
E A B C D F
G1 [1] [1] [1] [1] [1] [1]
G1’ [1] [-1] [-1] [-1] [1] [1]
G2
-1 0
0 1
 
 
 
1 0
0 1
 
 
 
1 - 31
2 - 3 -1
 
 
  
1 31
2 3 -1
 
 
  
-1 31
2 3 -1
 
 
  
-1 - 31
2 3 -1
 
 
  
3. The sum in (GOT) for G1’ and G2 vanishes:
1 0 -1 0 1 - 3 1 3 -1 3 -1 - 3 0 01 1 1 1
.
0 1 0 1 0 02 2 2 2- 3 -1 3 -1 - 3 -1 3 -1
            
                 
                   
Representation theory - orthogonality of irreducible representations
The irreducible representations of P(3):
E A B C D F
G1 [1] [1] [1] [1] [1] [1]
G1’ [1] [-1] [-1] [-1] [1] [1]
G2
-1 0
0 1
 
 
 
1 0
0 1
 
 
 
1 - 31
2 - 3 -1
 
 
  
1 31
2 3 -1
 
 
  
-1 31
2 3 -1
 
 
  
-1 - 31
2 3 -1
 
 
  
The sums in (GOT) for a=b are:
2 2 2 22 2
1 0 -1 0 1 - 3 1 3 -1 3 -1 - 3 3 01 1 1 1
.
0 1 0 1 0 34 4 4 4- 3 -1 3 -1 - 3 -1 3 -1
            
                 
                   
G1:
G1’:
G2:
             
2 2 2 2 2 2
1 -1 -1 -1 1 1 6 .     
             
2 2 2 2 2 2
1 1 1 1 1 1 6 .     
Representation theory - orthogonality of irreducible representations
The irreducible representations of P(3):
E A B C D F
G1 [1] [1] [1] [1] [1] [1]
G1’ [1] [-1] [-1] [-1] [1] [1]
G2
-1 0
0 1
 
 
 
1 0
0 1
 
 
 
1 - 31
2 - 3 -1
 
 
  
1 31
2 3 -1
 
 
  
-1 31
2 3 -1
 
 
  
-1 - 31
2 3 -1
 
 
  
The sums in (GOT) for a=b=G2 are:
2 2 2 22 2
1 0 -1 0 1 - 3 1 3 -1 3 -1 - 3 3 01 1 1 1
.
0 1 0 1 0 34 4 4 4- 3 -1 3 -1 - 3 -1 3 -1
            
                 
                   
2 2
1 1 1
( ) ( ) ,
n
rm j mu j rm mu ru
j m ma a
n n
D A D A
d d
  
  
  
G2:
Representation theory - orthogonality of irreducible representations
The relations of (GOT) can be interpreted in the following way. The matrix
elements of a unitary irreducible representation can be arranged in a set of
(row or column) vectors in the space of dimensionality n (order of the group):
, , 1 , 1 ,/ [ ( ), ( ),..., ( )].a rs a a rs a rs a rs nv d n D A D A D A
Individual vectors of this set are labeled by the symbol of the representation,
a, and by the indices r and s of the row and column, respectively. All these vectors
are mutually orthogonal (“projections on other members of the family vanish”).
In addition, the inclusion of the normalization factor of the square root ensures the
normalization (the square of the length of the vector is unity):
, , 1.a rs a rsv v

The maximum number of orthogonal vectors in the n--dimensional space is n.
Representation theory - orthogonality of irreducible representations
The irreducible representations G1, G1’:, and G2 of P(3) contain the following set
of orthonormal vectors:
normalization
factor
v1 v2 v3 v4 v5 v6
1/6 1 1 1 1 1 1
1/6 1 -1 -1 -1 1 1
1/3 1 -1 1/2 1/2 -1/2 -1/2
1 0 0 -1/2 1/2 1/2 -1/2
1 0 0 -1/2 1/2 -1/2 1/2
1/3 1 1 -1/2 -1/2 -1/2 -1/2
Representation theory - characters
The matrices (Da(Aj), j=1,...,n) of a given representation are very useful in
dealing with the group G; however, they suffer from the arbitrariness with regard
to similarity transformation , UDaU-1. Consequently, there is a reduction of
information content circumventing this arbitrariness.
Let ca(Aj) denote the trace of the matrix Da(Aj),
  ,
1
( ) Tr ( ) .
ad
a j a j a jj
j
A D A Dc

  
The set of the traces for all group elements,
1 2( ), ( ),..., ( ),a a a nA A Ac c c
is called the character of the representation (Da). Since
   Tr ( ) ( ) Tr ,jk kj kj jk
j k k j
UD U D D U DU     
the similarity transformation does not change the character of a representation:
         1 1 1 1
Tr Tr ( ) Tr ( ) Tr ( ) Tr .UDU U DU DU U D U U D   
   
Equivalent representations are related to each other by a similarity transformation
→ they have the same character.
Representation theory - characters
character
representation
c(E) c(A) c(B) c(C) c(D) c(F)
G1 1 1 1 1 1 1
G1’ 1 -1 -1 -1 1 1
G2 2 0 0 0 -1 -1
The group elements within a class are related by conjugation, the corresponding
representation matrices by a similarity transformation→
the corresponding entries in the character of any representation are the same.
Character table of the irreducible representations of P(3):
Its economized version (listing only the classes):
class
representation
C1=(E) C2=(A,B,C) C3=(D,F)
G1 1 1 1
G1’ 1 -1 1
G2 2 0 -1
Representation theory - characters
Characters of irreducible representations (Da(Aj), j=1,...,n) and (Db(Aj), j=1,...,n)
satisfy the orthogonality relations:
which follow easily from the great orthogonality theorem for the matrices. The
characters also satisfy the second orthogonality relation, containing a
sum over classes instead of the group elements:
*
1
( ) ( ) ,
n
a j b j ab
j
A A nc c 


*
( ) ( ) ,k k
a k b k ab
k
n n
C C
n n
c c 
where ca(Ck) is the (common) entry in the character of the k-th class, formed by nk
elements. The number of classes, nc, is equal to the number of inequivalent
irreducible representations , nk. This is the maximum number of mutually orthogonal
vectors in the nc-dimensional space.
Representation theory - characters
character
representation
c(E) c(A) c(B) c(C) c(D) c(F)
G1 1 1 1 1 1 1
G1’ 1 -1 -1 -1 1 1
G2 2 0 0 0 -1 -1
The orthogonality of the rows of the character table of the irreducible representations
of P(3) can be checked by inspection:
as well as the orthogonality of the columns of the table containing classes:
class
representation
C1=(E) C2=(A,B,C) C3=(D,F)
G1 1 1 1
G1’ 1 -1 1
G2 2 0 -1
Representation theory - constructing character tables
After finding classes, the following properties of representations are useful for the
construction of character tables:
(1) The number nr of inequivalent irreducible representations is equal to
the number of classes, nc=nr.
(2) The sum of squares of the dimensionalities, dj, j=1,..., nr , is equal to the order n of
the group.
(3) There is always one identity representation, showing up as a row of 1s.
(4) There is always one class composed of the identity element, showing up as a
column with the traces of unity matrices, i.e., the dimensionalities.
(5) The orthogonality of characters applies, both for rows and columns.
Representation theory - constructing character tables of P(3), isomorphic with C3v
using the Schoenfliess notation for symmetry operations of point groups
3
y
x
1
2
class: E 3sv 2C3
symmetry operations:
identity mirror plane 1
mirror plane 2
mirror plane 3
rotation by 2/3
rotation by 4/3
E: identity; needed in order to form a group.
Cn: rotation by 2/n; the rotation axis is called n-fold.
The axis of highest n, called principal, is “vertical”.
s: reflection in a plane, with three kinds of suffixes.
sh: reflection in a horizontal plane.
sv: reflection in a vertical plane.
sd: reflection in a vertical diagonal plane.
I: inversion, x↔-x, y↔-y, z↔-z.
Sn : improper rotation by 2/n, consisting of the
rotation by 2/n, folloved by a reflection in a
horizontal plane (shCn).
Representation theory - constructing character tables of P(3), isomorphic with C3v
class
representation
E 3sv 2C3
G1 1 1 1
G1’ 1
G2 2
The first row and column
are trivial:
The second row has to be
orthogonal to the first,
with the proper value of
the summed squares:
The second and third
columns have to be
orthogonal to the first:
done!
class
representation
E 3sv 2C3
G1 1 1 1
G1’ 1 -1 1
G2 2
class
representation
E 3sv 2C3
G1 1 1 1
G1’ 1 -1 1
G2 2 0 -1
A résumé of the symmetry group C3v (isomorphic with P(3))
using the example of the ammonia (NH3) molecule (N out of the plane of H3):
6 symmetry operations, , 6-dimensional space of orthogonal vectors forming the
characters
3 classes of conjugate elements, 3x3 table of characters
class
representation
E 3sv 2C3
G1 1 1 1
G1’ 1 -1 1
G2 2 0 -1