This package is dated 2008 and has never been redistributed with GAP. The file tst/ccgroup.tst has examples with IdGroup - are those reproducible? If not, it might be that GAP evolved since 2008 and HAPcocyclic needs to be updated. Or maybe there is a newer version, incorporated into another part of the HAP project? I hope the authors will be able to clarify this.

About 1c: GAP can tell you for a particular $q$. For small-sized groups it's not difficult to write a straightforward check, and with some clever use of theory you can go a bit further. GAP Software Carpentry Lesson may be of some help.

@AlexanderChervov GAP website does not provide online GAP calculator. In your case, I suggest to use cocalc.com which was formerly known as SageMathCloud. You will be able to run much longer calculations, not limited by 2 minutes like Magma online calculator, store your code for further runs, etc.

We need to distinguish web-interface and browser-based interface (to local GAP installation). It seems that you need the former one, while some answers suggest the latter. For installation, please contact GAP Support and tell which errors did you get. Finally, if you're not allowed to run exe-installer, get win-zip archive from gap-system.org/Releases - it does not require admin rights. You will have to manually edit bat-files to point to the installation directory.

To be precise, GAP Small Groups Library contains more groups than MAGMA: see gap-system.org/Packages/sgl.html. Also, some more $p$-groups appeared in the recently developed SglPPow package, see gap-system.org/Packages/sglppow.html

See also math.rwth-aachen.de/~Thomas.Breuer/atlasverify/index.html and in particular math.rwth-aachen.de/~Thomas.Breuer/atlasverify/overview.html

Have you managed to calculate this at the end? I think this experiment could make a good addition to my question at math.stackexchange.com/questions/1607517 ...

@NateEldredge: Having this experiment easily reproducible could make a very good answer to my question at math.stackexchange.com/questions/1607517 ...

@DerekHolt I think ATLAS could make one of the top answers to my question at math.stackexchange.com/questions/1607517 ...

Could you please add a definition for an interval of finite groups?

In case anyone willing to suggest new functionality or submit contributions, KBMAG package is now hosted on GitHub here: github.com/gap-packages/kbmag

@JustinBenfield the default for -o is 2g, so what you see in GAPInfo.CommandLineOptions could be just the default. Try different value to check that your changes actually have effect. In general, what you're doing to set -o should work. I have only VM with a smaller amount of memory - will make some changes when I will get access to the real machine.

@JustinBenfield: thanks, then I think you should be able to use 4g. How did you specify -o option? Have you edited .bat-file manually? Could you check that the shortcut in the start menu points to exactly that file?

I expected Windows in the reply... Have you compiled it yourself? (Please use @name to ping me in your reply, otherwise I am not notified. We may also continue by email.)

@JustinBenfield: regarding -o option, which operating system do you use?

@StefanKohl: thanks - yes, I think that is the crucial technique. Maybe converting to PC groups would also be fine, but by all means avoiding leaving them represented as groups of adthomorphisms.

@StefanKohl: I am also observing long runtime using GAP 4.7.9. Which version of GAP do you use?

You may be interested also in SmallGroupsInformation function - it gives some more details how the groups of a given order are sorted in the library.