We know the Lorentz group is $O(3,1)$ in 4 dimensional spacetime.
We know that there are 4 disconnected components in Lorentz group $O(3,1)$, and https://math.stackexchange.com/q/2204349/
$$\pi_0(\mathrm{O}(1,3)) \cong \mathbb{Z}_2\times\mathbb{Z}_2.$$
My question is that in QFT we have a discrete charge conjugation symmetry $C$, parity $P$ and time reversal $T$.
We know the $P$ and $T$ flips the 4 disconnected components of $O(3,1)$ into each other.
How about the charge conjugation $C$, does it sit in the Lorentz group? Or does it only act on the matter field? How do we understand $C$ in the Lorentz group $O(3,1)$?