Comment turned into answer.
You can easily buy a 6:1 impedance transformer to do the job. That gets you close -- indeed, for me and my equipment (ham radio that is) it is close enough. I bought a 6:1 transformer for about \$60 or \$70 from DXEngineering maybe 10 years ago. Have since sold it at a ham-fest.
I am assuming that your question is about amateur "ham" radio in making the following comments.
Along the output from your transmitting equipment (PA or whatever) you have multiple opportunities for a mis-match. This involves the transmission line connected to your equipment, anything in that path such as antenna couplers, and the antenna itself.
You want to do two things: (1) get the best match practical for the antenna and frequency of operation; and, (2) have the lowest practical loss path.
Usually any SWR under 2:1 meets the needs of (1). Above 2:1 and especially over 3:1 you definitely need some type of matching solution such as an antenna coupler, transmission line transformer, and so forth. Most modern amateur radio transceivers will fold back power levels when the SWR seen at its output is above 2:1.
For item (2) above, you want low-loss transmission line for most if not all of the path from transmitting equipment to antenna. For my wire antenna covering bands 80, 40, and 30 I use ladder line except for the last 15 feet. Ladder line is coupled to 50-ohm coax (LMR400) using a 4:1 transforming balun. The loss on the remaining coax is insignificant even when I operate at QRP levels (which I often do). I also use an antenna coupler/tuner (Elecraft KAT500).
For my mini-beam antenna (bands 20 - 10 meters) I use LMR400 coax for the entire path because the matching impedance with the antenna itself ranges between 1.1:1 and 1.8:1 (which I consider very good).
Note LMR400 coax is not the cheapest coaxial transmission line but it is low loss.
Very curious about your PA 420 ohm output impedance. In the last 55 years of ham radio activity, I have never seen any amateur radio equipment, even home brew, that did not have a 50-ohm output impedance. Again, I am assuming amateur radio as the OP question did not make it clear if it is something else.