I want to run the same slideshow, in one place by clicking on an image and in another place by clicking on an icon. How can I do that?

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Is there something I am missing? Or am I going about it the wrong way altogether?

You're going about it the wrong way altogether. It's much simpler than you think, thanks to the "showThis" option. Assign the same group string to each link and set "showThis:false" on the ones that you don't want to become part of the gallery set. Those links will still invoke the gallery.

This is done right at the top of the Floatbox demo page. The first row of thumbnails invokes a standard gallery set. The "Include Index Links" and "View as Slideshow" links immediately below that row invoke the same gallery with different options but don't become members of the gallery set because of the showThis:false setting assigned to them.

Brilliant! Thanks.