I'm reading Peter Webb's book on finite representation theory and after calculating the last row of the character table of a group, he mentions that it's the character table of $C_5\rtimes C_4$ and it is the only group that has this character table (it's an exercise to prove it).
I was thinking about how to prove it, but I can't even get started. I got a hint that I should think about Sylow subgroups. How does that connect to characters?
Thank you in advance!