The author of the above referred post is a friend of mine. The group theory data used there is based on the famous BC book, rather than being generated entirely from scratch.

I know nothing about Space Groups, but I did find this post. Not sure if it helps you.

If you have an Entity then you need to look at its "Properties" to see if it has something useful.

According to your above comment, I tried the following:

EntityProperties["CrystallographicSpaceGroup"]

{EntityProperty["CrystallographicSpaceGroup", "BravaisLattice"], 
 EntityProperty["CrystallographicSpaceGroup", "CrystalFamily"], 
 EntityProperty["CrystallographicSpaceGroup", 
  "CrystallographicPointGroup"], 
 EntityProperty["CrystallographicSpaceGroup", "CrystalSystems"], 
 EntityProperty["CrystallographicSpaceGroup", "EntityClasses"], 
 EntityProperty["CrystallographicSpaceGroup", 
  "HermannMauguinNotation"], 
 EntityProperty["CrystallographicSpaceGroup", "LatticeSystem"], 
 EntityProperty["CrystallographicSpaceGroup", "Name"]}

But I still don't know how to make further use of it.

It's very unclear to me what you want to do,

Many things can be done if I have the specific representation of a space group, such as computing the multiplication table, character table, and irreducible representation, etc.

or what you mean by building the space group by scratch.

I say this because I didn't know of any such entities at the time of posting this issue.

How to get the specific group elements that allow further manipulation of the group. For example, as shown below, given by the SpaceGroupIrep package:

Although the SpaceGroupIrep package has great application value for many problems in quantum mechanics and condensed matter physics. It is essentially a database-based tool, that is to say, it has not implemented the low-level group theory related algorithm. The space group information implemented in this package has its own definition format. As a result, it can not directly use the relevant group theory manipulation commands provided by wolfram language to facilitate the related study.