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In Georgi's book "Lie Algebras in Particle Physics", section 1.7, he presents a $2\times 2$ irreducible matrix representation of $\mathbb{S}_3$. But it is not mentioned how to derive it.

Given a representation of non-abelian permutation group $\mathbb{S}_3$ (or any other finite group), it is trivial to verify whether it satisfies the multiplication table or the group composition law.

But is there a rule by which one can explicitly construct irreducible representation of $\mathbb{S}_3$ (of dimension $>1$)? In particular, I want to construct a $2\times 2$, or a $3\times 3$ irreducible matrix representation of $\mathbb{S}_3$.

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    Yes, one can give an algorithm for finding all representations of the symmetric groups $S_n$ (as subrepresentations of the group algebra $\mathbb{C}[S_n]$). However, this strikes me as a pure math question which should be asked and answered at [math.SE]. – 2017-01-09
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    All irreducible representation are indexed by partitons of $n$ called specht modules. These are the only irreducible representation of symmetric group – 2017-01-09

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The irreducible representations of $S_3$ have dimensions $1$ (there are two of these, the trivial one and the sign representation) and a 2-dimensional representation obtained by identifying $S_3$ as the symmetry group of an equilateral triangle (the three cycles act by rotation matrices and the transpositions act as reflections).

There is a 3-dimensional representation of $S_3$ obtained by letting elements of $S_3$ act by permuting a basis $\{v_1,v_2,v_3\}$ (so elements of $S_3$ correspond to permutation matrices in this basis). This representation is not irreducible, though. the subspace spanned by $v_1+v_2+v_3$ is $S_3$-invariant (so gives a copy of the trivial representation). The compliment spanned by $\{v_1-v_2,v_2-v_3\}$ is irreducible and equivalent to the 2-dimensional representation described in the first paragraph.