See symmetric group:S3. We take the symmetric group on the set of size three.
Elements
See element structure of symmetric group:S3 for full details.
Element orders and conjugacy class structure
Review the conjugacy class structure:
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Partition 
Partition in grouped form 
Verbal description of cycle type 
Elements with the cycle type in cycle decomposition notation 
Elements with the cycle type in oneline notation 
Size of conjugacy class 
Formula for size 
Even or odd? If even, splits? If splits, real in alternating group? 
Element order 
Formula calculating element order

1 + 1 + 1 
1 (3 times) 
three fixed points 
 the identity element 
123 
1 

even; no 
1 

2 + 1 
2 (1 time), 1 (1 time) 
transposition in symmetric group:S3: one 2cycle, one fixed point 
, , 
213, 321, 132 
3 

odd 
2 

3 
3 (1 time) 
3cycle in symmetric group:S3: one 3cycle 
, 
231, 312 
2 

even; yes; no 
3 

Total (3 rows  3 being the number of unordered integer partitions of 3) 
 
 
 
 
6 (equals 3!, the size of the symmetric group) 
 
odd: 3 even;no: 1 even; yes; no: 2 
order 1: 1, order 2: 3, order 3: 2 


Multiplication, conjugacy and generating sets
Review the multiplication table in cycle decomposition notation:
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Review the multiplication table in oneline notation:
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Element 
123 
213 
132 
321 
231 
312

123 
123 
213 
132 
321 
231 
312

213 
213 
123 
231 
312 
132 
321

132 
132 
312 
123 
231 
321 
213

321 
321 
231 
312 
123 
213 
132

231 
231 
321 
213 
132 
312 
123

312 
312 
132 
321 
213 
123 
231

Conjugation and commutator operations
Review the conjugation operation:
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Review the commutator operation:
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Here, the two inputs are group elements , and the output is the commutator. We first give the table assuming the left definition of commutator: . Here, the row element is and the column element is . Note that :
The corresponding table with the right definition:
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Here is the information on the number of times each element occurs as a commutator:
Conjugacy class (indexing partition) 
Elements 
Number of occurrences of each as commutator 
Probability of each occurring as the commutator of elements picked uniformly at random 
Total number of occurrences as commutator 
Total probability 
Explanation

1 + 1 + 1 

18 
1/2 
18 
1/2 
See commuting fraction and its relationship with the number of conjugacy classes.

2 + 1 

0 
0 
0 
0 
Not in the derived subgroup.

3 

9 
1/4 
18 
1/2 

Subgroups
See subgroup structure of symmetric group:S3 for background information.
Basic stuff
Summary table on the structure of subgroups:
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Quick summary
Item 
Value

Number of subgroups 
6 Compared with : 1,2,6,30,156,1455,11300, 151221

Number of conjugacy classes of subgroups 
4 Compared with : 1,2,4,11,19,56,96,296,554,1593

Number of automorphism classes of subgroups 
4 Compared with : 1,2,4,11,19,37,96,296,554,1593

Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems 
2Sylow: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one 3Sylow: cyclic group:Z3, Sylow number is 1, fusion system is noninner fusion system for cyclic group:Z3

Hall subgroups 
Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups. Interestingly, all subgroups are Hall subgroups, because the order is a squarefree number

maximal subgroups 
maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3).

normal subgroups 
There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3.

Table classifying subgroups up to automorphisms
For more information on each automorphism type, follow the link.
View the lattice of subgroups as a picture:
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Linear representations
For background, see linear representation theory of symmetric group:S3.
View overall summary:
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Item 
Value

Degrees of irreducible representations over a splitting field (and in particular over ) 
1,1,2 maximum: 2, lcm: 2, number: 3 sum of squares: 6, quasirandom degree: 1

Schur index values of irreducible representations 
1,1,1

Smallest ring of realization for all irreducible representations (characteristic zero) 

Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) 
(hence, it is a rational representation group)

Condition for being a splitting field for this group 
Any field of characteristic not two or three is a splitting field. In particular, and are splitting fields.

Minimal splitting field in characteristic 
The prime field

Smallest size splitting field 
field:F5, i.e., the field of five elements.

View representations summary:
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Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.
View character table:
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Basic stuff
Up to equivalence, there are three irreducible representations of symmetric group:S3 in characteristic zero: the onedimensional trivial representation, the onedimensional sign representation (that sends every permutation to its sign), and the standard representation of symmetric group:S3, a twodimensional representation.