Let $G$ be a goup of order 144.
Roughly at 10:50 for No simple groups of order 66 or 144. - YouTube, he defines the homomorphism:
$$\psi : G \rightarrow S_4, \psi (g) = \sigma \in S_4 \text{ where $\sigma (i) = j$ if } g^{- 1} P_i g = P_j$$
Now why is this a homomorphism?
I suppose that $\psi (g) = \sigma_1, \psi (h) = \sigma_2$. Then $(g h)^{- 1} P_i (g h) = h^{- 1} (g^{- 1} P_i g) h = h^{- 1} P_{\sigma_1 (i)} h = P_{(\sigma_2 \cdot \sigma_1) (i)}$. So $\psi (g h) = \sigma_2 \cdot \sigma_1$. But I need $\psi (g h) = \sigma_1 \cdot \sigma_2$ for this to be a homomorphism?
What mistake did I make here?