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Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups.
By GAP, the set of all the sporadic numbers less than $2^{12}$ is $$L=\{266,506,759,1045,1288,1463,3795\}.$$

Question: What are the other sporadic numbers? Or at least, what are some next?

Motivation: If in your research you meet a group $G$ having a core-free maximal subgroup of index a sporadic number $n$, and if in addition you know that $G$ is neither $A_n$ nor $S_n$, then you can deduce immediately that $G$ is a sporadic simple group.

Sebastien Palcoux
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  • By sporadic group, do you mean sporadic simple group? Also, I don't think $2295$ should be on your list. – Derek Holt Jul 21 '19 at 21:51
  • I wonder why $22$ is not on this list. There are four primitive groups of degree $22$, $M_{22}$, $M_{22}.2$, $A_{22}$, and $S_{22}$. That suggests that you are not regarding $M_{22}.2$ as a sporadic group. But why not? – Derek Holt Jul 22 '19 at 07:16
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    It seems very unlikely that anyone will answer this question, but it might be useful to know that $492693551703971265784426771318116315247411200000000$ is an upper bound on the sporadic numbers. – Derek Holt Jul 22 '19 at 07:18
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    You can find an up-to-date list of maximal subgroups of sporadic groups here. This will tell you all of the potential sporadic numbers. Note that there is a slight query over the upper bound I gave in the previous comment. This was based on the maximal subgroup $41:40$ (order $1640$) of the Monster, but the possibility of a maximal subgroup $L_2(13)$ of order $1092$ has not been conclusively excluded. – Derek Holt Jul 22 '19 at 08:26
  • @DerekHolt: Yes, sporadic simple group. You are right, I misread my computation, it is not $2295$ but $1288$ (post just fixed). The primitive group of degree $22$ denoted $M_{22}:2$ is not simple. What is your favorite candidate for the next sporadic number? – Sebastien Palcoux Jul 22 '19 at 10:33
  • But why are you not regarding $M_{22}:2$ as a sporadic group? Or, more generally, please provide some motivation for this question. – Derek Holt Jul 22 '19 at 10:44
  • @DerekHolt: because it is not simple. – Sebastien Palcoux Jul 22 '19 at 10:45
  • Another question - do you really mean "all non-alternating and non-symmetric groups" or (which i think would make more sense) do you just want to exclude $A_n$ and $S_n$? – Derek Holt Jul 22 '19 at 10:46
  • @DerekHolt: Let $P_n$ be the set of primitive permutation groups of degree $n$ (upto isom.). What I mean is that $n$ is denoted a sporadic number if $P_n \setminus { A_n,S_n }$ is non empty and contains only simple sporadic groups. – Sebastien Palcoux Jul 22 '19 at 10:50
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    Can you give some minimal motivation for this question? I would find it more natural to allow extensions of sporadic groups, like $M_{22}:2$. – Derek Holt Jul 22 '19 at 10:53
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    $4180$ is the next sporadic number (but it would be $4125$ if you allowed $\mathrm{HS}:2$). – Derek Holt Jul 22 '19 at 11:02
  • @DerekHolt: You are an expert (I am not), so what you find natural is doubtless relevant. The outer automorphism group of a sporadic group has order at most $2$. What do you want to allow exactly (and why)? Index $2$ split extension? I suspect you to have in mind this notion of sporadic number since several years. Please, feel free to improve the post as you want. – Sebastien Palcoux Jul 22 '19 at 11:58
  • I don't claim that "sporadic group" has a standard meaning, but if you asked me whether variants of sporadic simple groups $S$ that are close to being simple, such as $S.2$ or central extensions of $S$ were sporadic groups, then I would be inclined to say yes. But to be honest, it doesn't really make a huge amount of difference whether you allow extensions $S.2$ - it would just mean that there were a few more sporadic numbers, which might be agood thing, because it would provide extra information. If you like, I could write some extended comments as an "answer" in the next couple of days. – Derek Holt Jul 22 '19 at 12:35

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I cannot give a complete answer to this question right now, but I believe that it would be possible to answer it by writing a moderate amount of computer code that made use of existing results in the literature.

I am interested in solving the more general problem:

Given an integer $n > 0$ describe the primitive permutation groups of degree $n$.

At the moment, the only routine way of answering this question is by using the GAP/Magma databases of primitive groups which currently go up to degree 4095, but should be extended to degree (at least) 8192 before long.

The barrier to extending this database further is that classifying the affine primitive groups of prime power degree $n=p^k$ (up to conjugacy in $S_n$) is equivalent to classifying the irreducible subgroups of ${\rm GL}(k,p)$ up to conjugacy, and that is computationally difficult, and is likely to remain an insuperable barrier to extending the lists beyond degree about 20000 in the foreseeable future.

But provided one is willing to accept "lots of affine groups" as part of the answer to my question above, then I believe it should be possible to answer it for much larger values of $n$. I would hope to be able to answer it for $n \le 492693551703971265784426771318116315247411200000000$ which, as I said in a comment, is (modulo a small number of uncertainties about the maximal subgroups of the Monster) the largest potential sporadic number.

By the O'Nan-Scott Theorem, the primitive permutation groups fall into a number of categories. These include groups of affine type, which we have agreed that we will not attempt to classify completely for large $n$. The second most frequently occurring type are the primitive permutation representations of degree $n$ of almost simple groups, which arise from maximal subgroups of almost simple groups of index $n$.

I believe that primitive groups of degree $n$ in the remaining O'Nan-Scott categories will be relatively easy to list, although I haven't thought about that in detail. They arise for relatively few values of $n$ - $n$ must either be a power of the order of as nonabelian simple group or a proper power of the degree of a smaller primitive group.

So the most difficult problem arises from the maximal subgroups of the almost groups $S$, and in fact results in the literature enable us to calculate those for very large $n$. To start with, the minimal degree $n$ of such maximal subgroups is known for all finite simple groups.

If $S = A_m$ or $S_m$, then we already know all maximal subgroups for $m \le 4095$ from the existing primitive groups database, and the intransitive and imprimitive maximals are easily described. For $m > 24$, a primitive maximal subgroup of $A_m$ or $S_m$ is known to have order at most $2^m$, so these will concern us only for $n \ge 4096!/2^{4096} \sim 3.5 \times 10^{11786}$.

Similar results apply to the classical simple groups based on Aschbacher's results about their maximal subgroups, which were made much more precise by Kleidman and Liebeck.

I have not yet checked what is known about the maximal subgroups of the exceptional groups of Lie type. These have been classified completely for the groups of small Lie rank, such as the Suzuki and Ree type groups, and for groups of larger rank, I believe that known results allow us to say that any unknown maximal subgroup has very large index indeed.

Finally the maximals of the sporadics and their extensions (all of degree at most 2) are all known apart from a very small number of uncertainties about the maximals of the Monster. It is currently unknown whether there are any such maximals with socle $L_2(13)$ or $L_2(16)$. (Unfortunately I am not sure whether anyone is currently attempting to resolve these remaining cases.)

I have computed a complete list of $285$ indexes of maximals of sporadic groups (ignoring the uncertainties for the Monster). Some of these also arise for degree two extensions, so they are not sporadic numbers according to your definition. Removing those from the list leaves 181, but some of the smaller numbers (such as $11$) are known not to be sporadic.

Derek Holt
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