We need to define a isomorphism :C(1,2) Sym(2)xSym(n-2). If is a permutation in C(1,2) then = f i.e. some permutation which fixes 1 and 2, or = (1,2)f. If =f then define ( ) = (id,f|A). If =(1,2)f then define ( ) = ((1,2),f|A).
To show that this subgroup is the whole of C_G(t), you need to show that any permutation that takes 1 or 2 to a number other than 1 or 2 is not in C_G(t). Suppose for example that σ is a permutation that takes 1 to 3. Then the product σ(12) takes 2 to 3. If σ(12) = (12)σ then (12)σ must take 2 to 3. This implies that σ takes 2 to 3. But that contradicts the fact that σ takes 1 to 3. The contradiction shows that σ is not in the centraliser of (12).