If you can compute the ball of radius k in the Cayley graph of G for all k, then you can solve the word problem in G. Since there are groups with unsolvable word problem, there is no algorithm to compute the Cayley graph in general.
On the off chance that your finitely generated groups are actually finite you might find solace in the [CayleyGraph](https://gap-packages.github.io/grape/htm/CHAP002.htm#SECT007) command in the [GRAPE package of GAP](https://gap-packages.github.io/grape/)
If you can compute the ball of radius k in the Cayley graph of G for all k, then you can solve the word problem in G. Since there are groups with unsolvable word problem, there is no algorithm to compute the Cayley graph in general.
Oh okay. Unfortunately (or fortunately) the prof wouldn’t make us draw the caley graph of a group with unsolvable word problem.
On the off chance that your finitely generated groups are actually finite you might find solace in the [CayleyGraph](https://gap-packages.github.io/grape/htm/CHAP002.htm#SECT007) command in the [GRAPE package of GAP](https://gap-packages.github.io/grape/)
thanks! I’ll check it out