What is the firing order?

Not sure where you found that firing order as that engine uses a simple 1-2-3-4-5-6 firing order. It uses a waste spark system to fire each coil, however when the spark happens one cylinder on the coil will be on the exhaust stroke while the other is coming up to fire on the power stroke. So, cylinders 2 and 5 actually fire at the same time as do 1 and 4 and 3 and 6 as those are the way they are paired. As such you can swap the wires on the 2-5 coil and it won't matter, same with the 1-4 and the 3-6. The firing order itself is set in and by the computer.

That is what my coil pack is numbered 412536.

That isn't the firing order. Its just which cylinders are paired. 4 and 1 use the same coil, you can reverse those wires and it would still do the same thing. The actual firing order though is 1-2-3-4-5-6. If you were to break the system down it actually only has three control wires. one wire triggers each coil and each coil splits to two cylinders. So, the 41 is a single coil for cylinders 1 and 4 and fires both at the same time. 25 is another coil and 36 is the last one. But that isn't the firing order, just the cylinder numbers to route the wires.

My firing order on my coil pack is 412563 and my car ran good. So, why is the book saying 145263 with the firing order 123456? my car doesn't even come close what the manual says. anyone know why?

Hello, your vehicle has what is called a waste spark system, meaning cylinders 1,4 -2,5 -3,6 fire the cylinders at the same time. I'll post a diagram of the coil pack set up. But for example, as cylinder 1 is on its compression stroke and 4 is on its exhaust stroke the 1 coil fires both cylinders. The spark travels through the number 1 spark plug, through the engine block, back out to the number 4 spark plug and returns to the coil and is dissipated. They are companion cylinders. Each coil fires 2 cylinders.
The correct firing order is 1,2,3,4,5,6.
This is from All Data.

No my firing order on the coil pack is 412563.

What's the issue with the vehicle? Service info only has that firing order.