Preface
I'll preface this by saying I asked this question at the Mathematica StackExchange yesterday (https://mathematica.stackexchange.com/questions/282024/bliss-automorphisms-how-to-list-all-automorphisms-using-the-output-generators) and received a good answer for listing automorphisms using Mathematica software. The software uses a function called "GroupElements" that lists the automorphisms. However, what I'm basically asking here is how a function like "GroupElements" works under the hood or a general method so I could apply it to any software that outputs generators (Bliss, Nauty, etc.).
Question
I am using the program called Bliss (http://www.tcs.hut.fi/Software/bliss/index.html) to try to extract all automorphisms of an input graph. The software itself does not list all of the automorphisms directly but instead outputs a list of "generators" that are supposed to be used to list all automorphisms by combining the output of the generators.
What I'm stuck on is that I don't see a clear pattern of how to list ALL automorphisms using the generators. Bliss may output two generators, but the total number of automorphisms could be 4 (which includes the identity mapping).
How would one combine the output of the generators to list ALL autormorphisms? The generators could come from Bliss or similar software that outputs generators.
Example 1
Source: Page 24 http://www.cs.columbia.edu/~cs4203/files/GT-Lec2.pdf
Bliss Output:
Graph read in 0.00 seconds
Initial partition computed in 0.00 seconds
Generator: (1,8)(2,7)
Level 1: orbits=6, index=2/2, all_same_level=1
Generator: (3,5)(4,6)
Level 0: orbits=4, index=2/2, all_same_level=0
Nodes: 5
Leaf nodes: 3
Bad nodes: 0
Canrep updates: 1
Generators: 2
Max level: 2
|Aut|: 4
Total time: 0.00 seconds
The output says |Aut| = 4, which is correct since there are 4 automorphisms per the example screenshot.
The output has the generators that correspond to two of the automorphisms from the example. Another automorphism is the identity. But what I don't know how to get from these generators is the last automorphism shown in the screenshot using the generators.
Example 2
As another example from a very similar tool to Bliss called Nauty (https://users.cecs.anu.edu.au/~bdm/nauty/), Nauty also lists the generators but not all automorphisms as explained in page 3 here https://users.cecs.anu.edu.au/~bdm/nauty/nug26.pdf with a good example. I'll place a screenshot below in case the link becomes dead one day. Here, there are 8 automorphisms that could be generated by just two generators.
Thanks, any help is appreciated.