# Given a finite group, how to figure out which chiral algebra can realise this symmetry?

The classification of the "minimal models" of chiral algebra gives us rational conformal field theories in two dimensions. For example, the classification of unitary representations of Virasoro algebra gives us minimal models, which have $$Z_2$$ global symmetry. Similarly, the minimal models of $$W_3$$ algebra all have $$S_3$$ symmetry, the leading member of this family corresponds to the critical 3-state Potts model.

Suppose I construct a statistical physics lattice model in two dimensions, which preserves a certain finite group symmetry, one natural question is whether this lattice model, in the thermodynamic limit, will go through a second-order phase transition as we change temperature. (It is not difficult to construct such lattice models, for example, one introduces a spin degree of freedom on the site of a square lattice, which transforms in a certain irrep of the finite group, one can then use the invariant tensor of this finite group to introduce coupling among neighboring spins).

To answer this question, it seems to me that the first question to ask is whether there is a 2d CFT whose fusion rule preserves such a finite group symmetry. This lead to the question in the title "Given a finite group, how to figure out which chiral algebra can realize this symmetry?" (Will coset construction give us some hint?)

The global symmetries you cite are properties of specific models, not of chiral algebras. With a Virasoro algebra, you could have no global symmetry (Lee-Yang minimal model) or $$S_Q$$ ($$Q$$-state Potts model) or $$O(N)$$, etc.
Even if your chiral algebra has a nontrivial group of automorphism, for example $$S_N$$ for an algebra made of $$N$$ copies of Virasoro, this could in principle be broken in a specific model.
If you start with the $$2d$$ Ising model on the lattice, you notice the $$\mathbb{Z}_2$$ symmetry, then you argue that the critical limit exists, and then you start looking for a CFT with Virasoro and $$\mathbb{Z}_2$$ symmetries. The minimal model $$M(4, 3)$$ has the required symmetries, but it may not be what you want: if you are interested in nonlocal objects such as cluster connectivities, you need a bigger CFT.