fagan, n. — Green’s Dictionary of Slang

fagan n.

also fagin

[the character Fagin from Charles Dickens’s Oliver Twist (1837–9); despite Fagin’s poss. homosexual paedophilia and the use of ‘fag’, no specific gay context is implied]

the penis.

1958		(con. 1940s) B. Behan Borstal Boy 89: They’d be there till the Lord would call them before he’d get down to introducing fagan.
1979		J. Morrow Confessions of Proinsias O’Toole 36: I’ve never had much time for social intercourse with any woman who has neither Fagan or a frying-pan in her hand.

In phrases

bury old Fagin (v.)

of a man, to have sexual intercourse.

1984

Partridge DSUE (8th edn) 161/2: C.20.

introduce her to Fagan (v.)

of a man, to have sexual intercourse.

2000

Roger’s Profanisaurus 🌐 Fagan n. Penis. As in “How did you get on last night. Did you introduce her to Fagan?”.