meaning of the phrase "make a show of something"

[Klpno]

Hi everyone.
I'd like to ask about the meaning of the phrase "make a show of something".

Some soldiers are sitting in the bar in the Officers' Club. One of the soldiers is surprised by this.

Making a show of looking at his sleeve, Zipes said? "Well, I ain't no officer. Neither ain't none'a you. Sergeants over there, "he pointed at Goodier and Gnaizda, who were sitting at a table talking with a corporal Vega didn't know," they officers, maybe, but we ain't. So why we in here?"

Does the phrase "Making a show of looking at his sleeve" mean that he is looking at his sleeve in a very exaggerated way or does it mean that he pretends that he's looking at it?

[Conchita]

[Yankee]

Hi KLPNO

I don't think "pretend" fits the context very well here. He didn't pretend to look, he actually looked.

When you make a show of doing something, you do it in a very conspicuous/obvious/ostentatious manner.
.

[Klpno]

[Klpno]