What happens when a symmetry element is added or removed from a point group?
Inversion $i$ is equivalent to a two-fold improper rotation $S_2$. Introducing a $2n$-fold improper rotation axis to the existing $C_n$ point group will result in a new $S_{2n}$ point group, e.g. in your case $C_3 ⊗ i = S_6$.