# For which $n$ in $2048 < n < 4096$ is the exact number of groups of order $n$ up to isomorphism currently unknown?

The number of groups of order $$2^{11} = 2048$$ up to isomorphism is currently unknown, that is at least, as of 2023. The same can be said of $$2^{12}=2 \cdot 2048 = 4096$$. What about the number of groups of order $$n$$ for $$2048 < n < 4096$$?

Perhaps for example $$3072 = 1024 \cdot 3 = 2^{10} \cdot 3$$, as it has the same number of prime factors (with multiplicity) as $$2048$$? Or is this one known?

• Here you can find at least a criteria for orders of $n$ for which the number of groups is known: groupprops.subwiki.org/wiki/Number_of_groups_of_given_order Nov 19, 2023 at 17:17
• It is a very long time ago I dealt with groups , but if I remember it right , $3072$ is a hot candidate for an unsolved case. Nov 19, 2023 at 17:56
• For a list of the orders that are not currently in the SmallGroups database, go to magma.maths.usyd.edu.au/calc and run "for n in [2049..4095] do if NumberOfSmallGroups(n) eq 0 then n; end if; end for;" One could argue this is a candiate list for "unknown" orders. Of course, not all these orders are equally difficult, and some of them could be done without too much effort (for example, I am aware of an algorithm currently being implemented to deal with cubefree orders). Nov 19, 2023 at 21:35
• A more complete query can be done via groups.quendi.de, e.g. groups.quendi.de/?order_min=2049&order_max=4095 for the given range. Or use GAP gap-system.org which is free and open source. The cubefree orders up to 50000 are also already all in GAP. Nov 19, 2023 at 23:43

tl;dr: the only "unknown" order is 3072.

In the small groups library (available in the computer algebra systems GAP and Magma), many of the groups of order in the range from 2049 to 4095 are already available. Indeed, all but 216 of them. Here is a check in GAP that counts how many orders are missing

gap> Length(Filtered([2049..4095], n -> not SmallGroupsAvailable(n)));
216


The relatively new GAP package SotGrps by Eileen Pan (as far as I know equivalent code is also in Magma) adds support for more infinite families of groups, but in this case, that just takes care of 2 of the "missing" orders.

However, using e.g. the GrpConst package by Besche and Eick (with some minor optimizations by myself) plus enough patience and waiting time one can construct the groups of many of the missing orders. We did so as part of our paper

Eick, Bettina; Horn, Max; Hulpke, Alexander, Constructing groups of ‘small’ order: recent results and open problems, Böckle, Gebhard (ed.) et al., Algorithmic and experimental methods in algebra, geometry, and number theory. Cham: Springer (ISBN 978-3-319-70565-1/hbk; 978-3-319-70566-8/ebook). 199-211 (2017). ZBL1400.20014.

Now, this leaves some "hard" orders. In the end, just two are left, 2304 and 3072.

For the order $$2304=2^8\cdot 3^2$$, we were able to deal with it with a new algorithms presented in

Eick, Bettina; Horn, Max, The construction of finite solvable groups revisited, J. Algebra 408 (2014), 166–182, also available on the arXiv.

This only leaves order $$3072=2^{10}\cdot 3$$, which is currently out of reach. To resolve it, we'd first have to actually compute all the groups of order 1024; we know how many there are up to isomorphism ($$49\,487\,367\,289$$) and in principle have the tools to compute them, it's a matter of patience (CPU time, and perhaps RAM size etc.). But then you need to perform computations for each of these groups to get the "descendants" of order 3072. That would be a massive computations -- it might just be feasible if you have a big enough compute cluster and lots of time and write some improved code, but frankly, I am not sure it'd be worth the effort. That said, others might think the same about order 2304, so ....

You can browse a lot of the data we produced in our papers at https://groups.quendi.de.

Edit: Max Horn has just posted a better answer on this question with the same info as their posted comments on this answer as well as more details.

A partial answer to this question. This answer by Max Horn who developed a certain algorithm to enumerate group orders writes

This [the algorithm] is part of an on-going project to catalogue all groups up to order 10,000 (with a few orders excepted, e.g. multiples of 1024, as there are simply to many of these). So in particular, we skip groups of order 3072. There are already $$49,487,367,289$$ groups of order 1024, and I expect the number of groups of order 3072 to be several orders of magnitude larger.

The answer was posted in 2016 and I am struggling to find anything more recent (or any academic source, although as an undergraduate I'm not really experienced with that), so it seems that the number of groups of order $$3072$$ was unknown then in 2016 and is still unknown.

Other comments on that post also suggest that $$3072$$ is unknown.

• Yes, 3072 is unknown. Computing it would require iterating over each of the $49\,487\,367\,289$ groups of order 1024 and computing among other things the automorphism groups of all. As a first step, we would have to compute those groups, though (we currently only have the number). I guess In principle it would be possible to do it with a big enough cluster and if one is willing to wait long enough, but honestly I don't think it'd be worthwhile. Nov 19, 2023 at 23:40
• This query also confirms that 3072 is the only number in the given range were we don't know the answer: groups.quendi.de/?order_min=2049&order_max=4095 Nov 19, 2023 at 23:41
• As to an academic source, see also our paper "Constructing groups of ‘small’ order: Recent results and open problems" (joint with Bettina Eick and Alexander Hulpke), dx.doi.org/10.1007/978-3-319-70566-8_8 Nov 19, 2023 at 23:44
• @MaxHorn Thank you very much - this completely answers my question. If you'd like to write an answer about this I'd happily accept it, otherwise I'll edit this answer of mine with the result and say "see comments below from Max Horn" :) Nov 20, 2023 at 0:04