Love this song.
"Come On Eileen" may have been the only "hit" from Dexy's- I think it was used in a John Hughes movie or something -but that whole album (
Too-Rye-Ay) is fantastic.
Likewise, Men At Work ("Who Can It Be Now?"/"Down Under") got thought of as something of a novelty band, but that whole album is quite good.
Here's another example: The Hooters ("All You Zombies"/"And We Danced").
Nervous Night was the album, again- really good stuff. The leader of that band wrote and produced several hits for Cyndi Lauper.