Super Saiyan God deserved better. It's almost never used, and when it is used it's quickly forgotten about. That being said, it's probably the highest power jump from one transformation to the other. SSJ3 is nothing compared to SSJ God. Lets talk multipliers.
As far as I'm aware, we don't have an official Super Saiyan God Multiplier. Having said that, we can find a close enough estimate from what the show and manga give us. The easiest way for me is to compare Super Saiyan God to Potara Fusion.
In the anime, Goku says not even fusion could defeat Beerus, but this doesn't help us because Super Saiyan God wasn't enough to defeat Beerus. But this statement still helps us out a bit. Had Goku decided to fuse with Vegeta, he probably would have gone SSJ3 to become as strong as possible. Goku assumes a SSJ3 Vegito would be weaker than this potential Super Saiyan God form.
So how strong is a SSJ3 Vegito? Well first we need the Potara multiplier. I prefer the A + B times 100 = C equation, because according to Vados, "The new body and personality born from a Potara fusion.... are more than the sum of their parts and their power increases tens of times.". By tens of times, I'm assuming she means anywhere from 20 times to 190 times, so I'll go down the middle.
Lets assume that BoG Goku and Vegeta are both 1 to keep the numbers small. Base BoG Vegito would be 200 times stronger than either Goku or Vegeta, and a SSJ3 BoG Vegito would be 80,000 times stronger than base Goku or Vegeta, and Goku thinks SSJ God is stronger than that. Had Goku felt SSJ3 Vegito stood a better change against Beerus, he would have told Beerus. So we can assume that SSJ God is at least 1.25 times stronger than SSJ3 Vegito.
So the Super Saiyan God Multiplier is at least a 100,000 times multiplier, 250 times stronger than SSJ3, and 500 times stronger than base Vegito. Some people would bring up Gokus fight with Kefla, how Base Kefla was beating a Super Saiyan God Goku, but remember that Goku was still recovering from using UI for the first time.
It's likely that he couldn't use Super Saiyan God at full strength, the same way Vegeta couldn't use Super Saiyan Blue at full strength during the Universe 6 arc. There is the off chance that Kefla is just stronger than Vegito, but that's a discussion for a different day.
TLDR: Super Saiyan God is a 100,000 times multiplier.
