For background, see linear representation theory of symmetric group:S3.
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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.

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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.
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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.