wikipedia: probably more correct than whatever I say here

Ok so first, a reminder. Here is what the cyclic group looks like. It’s the group for addition modulo 5. So the little points are the group elements and the arrows represent action of a generator \((+1)\) on them.

OK, now let’s look at the direct product.

hmmm. I think this is a fine picture. Maybe this is a better picture.

But you get the idea, you basically can move via two clocks at the same time.

semi-direct products

Now what if we did a weird permutation in between?

so in this picture the outer-most square and the inner-most square are actually the same, but I didn’t want to draw too-dense arrows.

So what’s the deal with this weird product thing? Well. I dunno. It seems kinda cool though.