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OEIS shows the number of groups of order $n$ upto $2047$. The Magma-online-calculator uses a database, but already for $1024,2004,2016,...$ it cannot determine the number of groups. Maple seems to calculate the number of groups (unless it is too large), but unfortunately, I do not have access to maple.

Does anyone know an online-calculator for the number-of-group-function, or a table with the known numbers upto $10,000$ or more ?

Or alternatively, allows PARI/GP, at least in principle, to calculate the number of groups of order $n$ ? I programmed the case $n$ squarefree, but I have no clue how to manage arbitary numbers $n$.

Community
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Peter
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  • There isn't any general formula or algorithm known for computing the number of groups of a given order $n$, other than to construct all of them. Maple uses formulas for certain special cases, depending on the factorisation of $n$ (for instance, for square-free $n$, or for small powers of primes, etc.) and uses a table of known values for small $n<50000$. I expect GAP and Magma have something similar, but I don't know that PARI has that kind of functionality. – James Dec 01 '15 at 14:27
  • @James I was just curious. By the way, in the answer below, it is claimed that the number can be calculated in the case $p^2 q^2$. Do you know the formula ? – Peter Dec 01 '15 at 17:01
  • It's my understanding that some sort of formula or description is known, but I do not know it! I've never been able to get my hands on the papers where it is described. – James Dec 01 '15 at 20:33
  • For four primes, the situation is as follows.

    The case $p^4$ and the squarefree case are well-known.

    The case $p^2qr$ was done by Glenn (1906) http://www.ams.org/journals/tran/1906-007-01/S0002-9947-1906-1500737-3/S0002-9947-1906-1500737-3.pdf

    As far as I can see, the number of isomorphism type is not explicitly stated, but perhaps could be extracted with a little more work.

     – verret Dec 02 '15 at 06:24
  • The case $p^3q$ was done by Western (1898) http://plms.oxfordjournals.org/content/s1-30/1/209

    His count is at the end. 15 groups when $q=2$. 6 or 19 groups when $q=3$ and $p$ odd. (For $p=2$, there are $15$.) etc... Take all of this with a grain of salt, for example, according to magma, there are $7$ groups of order $3\cdot 5^3$. Moreover, his count in the table is also off for 189 and 351.

     – verret Dec 02 '15 at 06:24
  • As for the case $p^2q^2$, there is -Cheissin (1940) in Russian http://www.mathnet.ru/links/60d16d15641beeb3d01618e6fcc20a64/im3913.pdf -CHEN Songliang (2009) (in Chinese) http://en.cnki.com.cn/Article_en/CJFDTOTAL-HZSZ200904002.htm

    It has an English abstract and claims the following formula that, assuming $p$ and $q$ are odd, there are either 4,6,7,11, or 15 groups

    This also seems wrong, according to magma, there are 13 groups of order $441=3^2\cdot 7^2$, whereas he predicts $7$.
    Last reference: Lin, Huei Lung On groups of order $p^2q$, $p^2q^2$. Tamkang J. Math. 5 (1974), 167–190.

     – verret Dec 02 '15 at 06:25
  • I suppose the takeaway is that, while these results are within reach in principle, some of the computations are a bit delicate, especially when trying to count the classes. (Rather than just give a complete but possibly redundant list, which is much easier.) – verret Dec 02 '15 at 06:25

2 Answers2

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For the list of $n$'s which are included in the SmallGroups library, see : https://magma.maths.usyd.edu.au/magma/handbook/text/727

There are many missing n's which could be computed, even by hand (when $n=p^2q^2$, for example), but there are definitely some numbers less than $10000$ that are out of reach.

For example, the number of groups of order $2048$ is not actually known, higher powers of $2$ even less so, see for example https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf

(This is a good reference for your question in general.)

verret
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No-one has as of yet linked to Max Horn’s groups.quendi.de which answers your question and more. It contains the number of groups of order $n$ up to isomorphism for $1 \leq n \leq 50\,000$, where that is known.

It also gives information regarding the number of solvable groups and the number of nilpotent groups for the orders.

Robin
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