In order to prove that L, R and C are permutation representations of G on X = G, we proceed in two steps.

The maps L_g, R_g, C_g are bijections, so they belong to Sym(G).

The maps L, R, C are morphisms G -> Sym(G), so they are permutation representations.

We need to verify that, for each g, h G,

L_gh = L_g L_h.

This is indeed the case as, for each x G,

L_{g h}(x) = (gh)x = g(hx) = L_g(L_h(x)) = (L_g L_h)(x).

The proofs for R and C are similar.