Talk:Sporadic group

1905: 84: 74: 53: 1533:. I'll try and find a source for this analysis somewhere, and link it here. I will, for the sake of completion, seek other simple maximal groups that might be in common and list them here, after realizing I had forgotten about isomorphisms which could also be simple (but not finite of Lie type or alternating), of which there shouldn't be that many. This is all in an effort to bring clarity to the structure of the sporadic groups, which readers might find interesting to read here. 1691:(so to speak, as a zeroth term) in its family of groups -- and importantly so, because this means that there is a slight "sporadic" expression in this family of groups, and sporadic in the sense that it is not generated uniformly from a simple expression that continues to generate groups of the like (i.e. a family) -- so it is not in violation of this, in pure, "strict" terms. The following terms in the set, however are of Lie type, without exception. 22: 1739:(Although many, many, many infinite systematic families of finite simple groups are of Lie type, this is NOT a defining attribute, because the groups of prime order and the alternating groups also are not of Lie type. But both are infinite systematic families, with the consequence that they do not contain a sporadic element.) The family of commutator groups 1078:, Suz and O'N), none is a subquotient of the other. I'll keep looking a bit more when I come back from work, but I think this is the only alternating group which is shared between groups that are not subquotients within each other. It's actually special! We can wonder what we would do with this; the only maximal subgroup inside M that holds A 1631:

To my knowledge, a "link" from one article to a second does NOT have the meaning: The object in the first article IS of the type of the objects in the linked article (the link is only to be understood in a way similar to "see also".) So in my opinion, you have to look for a source for the sporadicity

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I want to point out that we could have answered the original question on the onset, in the sense that sporadic groups will only have simple sporadic subquotients (for the ones that do); there cannot exist a Lie group or any other simple group which would have inside it a sporadic group, simply by the

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In representations on the topic of sporadic groups for non-experts (Knowledge, MathWorld, Ronan) one looks in vain for an explanatory argument (preferably of a heuristic nature) which motivates the fact that there are only finitely many of them. Why, for example, is there no exceptional simple finite

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There is a discrepancy between this page and the one dedicated to the classification effort in general. This page refers to the sporadic groups as well as eighteen countably infinite families. The classification page, however, says there are only three such families necessary for the proof. Which

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I'll link the sources; it's a longstanding understanding that this group specifically has characteristics from both, and many-a-times it has been cited as being in an ambiguous classification, which is why it is classified as such in the literature (as either, both, or neither), from primary and

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Worth noting, in line with the showing of finite simple Lie groups that exist within the structure of maximal subgroups (of which four I found, and listed above), there are many alternating groups within the structure of different maximal subgroups of sporadic groups. It would be interesting to

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That is what the reference claims, we have to check its work first. The work should be correct, since we are taking into account the minimal field over all its elements, not just some of its elements. (Over some of its elements it is a stabilizer, as you pointed.) Altogether, the minimal

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I think it's great that we are discussing this, it's not obvious and maybe hard to find within the literature, so this is good to have. (I moved one more group inside another sub-bullet above, just in case). To answer your question, there is no other sporadic subquotient inside

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First and foremost, the character table in the ATLAS of Finite Groups shows 2 26-dimensional representations, exchanged by an outer automorphism, so the minimal faithful character should be 26. The online ATLAS of Finite Group Representations includes these representations

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You noted that there is a non-simple (finite) group of which all sporadic groups are subgroups (and thus subquotients). Hence, there exists a minimal such group; but that minimal group (or those minimal groups) does not have to be simple, has it? I think that

1887:. Because most of the literature does acknowledge this generalization of T (really, read into the actual papers), the way it is presented now is PROPER, and ACCURATE. Those references, as I have done for several in our (everyone's) article, I will find, 1423:

All of these exist relatively independently of one another and do not generate sporadic subquotients (naturally, by definition). Interestingly, they include members that are for the most part the smallest or smaller members of their respective classes.

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This is a total of three groups of Lie type (that I know of) that are found inside sporadic groups (when counting T as well), of which none are shared between any of the sporadic groups in whole. Technically, only two groups represent simple Lie groups

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example; that is outside the general framework of simple groups that would be shared between sporadic groups from groups that have sporadic subquotients (that are not covers of sporadic groups either), which actually holds some meaning:

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Also, Mathworld is mainly useful for finding summarized information, and needed sources of interest; it is not a primary source. We should not rely on it as a primary source. Part of the sporadicity of T is found in its

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and O'N actually hold two copies of these groups which are fused together, while Suz only holds one of these groups. Note too that, of the duplicate alternating groups that are isomorphic with Classical Chevalley groups

1548:"there cannot exist a Lie group or any other simple group which would have inside it a sporadic group, simply by the definition of how they are defined" ...but every group is a subgroup of an alternating group 616:

is involved within two different sporadic groups, one of which is a pariah. The only other possible simple subquotients would need to be alternating, some of which are congruent with Classical Chevalley groups

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Also, if someone wants to edit the diagram to include T then it fits under Fi22 and Ru. (I think there is no need to add a direct link under B. For example, there is no direct link from M22 to M24.) --

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nor from any other group of Lie type, so exceptional within the family of groups of Lie type, and a group like it does not exist inside families of other simple groups, i.e. cyclic or alternating

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Thank you so much for your work! As you remark: the non-sporadic Tits group is subquotient of the Rudvalis group. But "in between" means that there would also have to be a sporadic subquotient

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Regarding the bit from Mathworld, I did not want to remove it either because it is not incorrect, though not the best source either (to use as an example in this instance), naturally.

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are the only two such groups. It seems noteworthy since through them, they generate a further (more subtle) link between main generation groups and the pariahs. Also, within A

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is the middle indexed by order in this second generation, with a largest prime factor of 13 dividing its order as well, which makes it the only other sporadic group with

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appear). I'm going to take a look later, in the meantime maybe also seek that reference we want. o/ 07:52, 4 March 2023 (UTC), 00:17, 5 March 2023 (UTC); last updated

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So I investigated how many Lie groups exist within sporadic groups, in whichever algebraic structure, and found the following result (after a multitude of re-edits):

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where also 7 is the first prime number that does not divide its order; any number greater in proportion to 7 (the first being 14, the first integer greater than 13)

827:! It is unsurprising, though very pleasant to work out; this surely has been said somewhere and maybe listed in similar fashion, it's just a matter of finding it. 292:

Of course, it is easy to establish that there is no sporadic subquotient in between. But it is not obvious (although almost 100 % probable) that there is no other

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I was thinking of organizing them by date of discovery, to bring some novelty w/o OR. Else, by type or order would be appropriate too, as you mention, Nomen4Omen.

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sharing some properties of the sporadic groups (more than 1 or 2 maximal subgroups, has multiple semi-presentations, embeds in at least one important group (Fi

761:. Mainly because it is defined within the family of Lie groups, so by definition, the Tits group is organized by what would be generated from a Ree group F 2634: 130: 1467: 2342:

in turn) is of great consequence. Worth mentioning, 744 is the number of partitions of 7 into prime parts, where 744 is the constant term in the

1867:*while it has a small outer-automorphism (2, as with many sporadic groups), and is also one of seven classes of simple N-groups (which includes M 1003:— does not hold a duplicate isomorphism. Also, an aside, is that 2520 is the first number divisible by the first ten integers 1 through 10. Its 2242: 106: 2629: 404:

Another group in this generation contains a simple group of Lie type in its algebraic structure (of a maximal subgroup), but not as a whole:

1875:, Suzuki groups, and other non-simple groups); this shows that its Lie nature blends with its sporadic nature, from some examples at least* 1672:

Moreover, there has to be given an extremely close look on the attributes defining "sporadic": A finite simple group is called sporadic if

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Since the "Table of the sporadic group orders (w/ Tits group)" is sortable (for several columns), this issue should be solved in principle.

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simple factorized order, that is in the like of the other small sporadic groups, yet not from the same type of sporadicity, so to speak

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We can check common groups they'd share, this should be easy; there could also be some alternating groups in common that are simple.

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within a maximal subgroup) common between sporadic groups where one is a not subquotient of another. I checked and double checked, A

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that show its nature blends into both the realm of Lie groups and Sporadic groups; these might be some of the more prominent ones.

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quantify these too, and see any common subgroups they'd generate, in particular in the interest of the pariahs, where for example J

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This example, of course, aside from the subquotients inside a sporadic group that holds other sporadic groups inside. For example,

2064:) in the classification scheme, so to speak, which gives some reasoning as to why it is of import on the side of sporadic groups: 1755:

of finite simple groups. And the Tits group is a member of this infinite systematic family (and thus by definition not sporadic).

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A more basic fact is that it does not fit the order formula. Every finite simple group has order: a prime number, n!/2 for n: -->

2214: 1844:"The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but 1200: 1007:

is 576, which is the square of 24 (incidentally, the sum of its factors 2 × 3 × 5 × 7 is 24 as well; where 24 comes to play in

1019:. Other places where factors of 10 come into play inside sporadic groups would be: the irreducible complex representation of M 680: 1045:, and the irreducible complex dimensional representation of Ly (a pariah) and Th (3rd gen.), respectively, are 2480 and 248. 1973: 2018:

prime numbers, when counting powers separately (a nice association with a count of classes of Lie groups, when including

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While the alternating group on 7 letters is part of this grouping, the alternating groups on 5, 6, 8 and 9 letters are

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The table says that the minimal faithful character of the Tits group is 104, but there are multiple issues with this:

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factorization contains a total of four distinct primes (a low number, and minimum amongst the sporadics), otherwise

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Three groups in this generation contain simple groups of Lie type in their algebraic structure, but not as a whole:

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sporadic groups; see end-comment below of the same day as today... so this result has surely been stated prior.

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also contains 14 (twice seven) maximal subgroups (two are fused), alongside 69 conjugacy classes (thrice 23)

1456:(2), with no other simple maximal groups inside either. I misread actually, and sought simple groups shared 528:

One group in this class contains a simple group of Lie type in its algebraic structure, but not as a whole:

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is most unconnected with the rest of the sporadics; that really is a key here (it might boil down to which

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The diagram showing the subquotient relations contains the remark "with no simple subquotient in between".

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as third largest sporadic group, has an order divisible by 7). The minimal faithful Brauer character of

2264: 2262:= 7 dividing its order, the smallest maximum prime value dividing the order of any sporadic group), and 1792: 1391: 1298: 920: 765:(2), none of which have sporadic subquotients (like in general with any of the Lie groups); except for F 390: 232: 39: 83: 2581: 2570: 2566: 2529: 2525: 2126:, which also contains 13 as its largest prime factor dividing its order, with generators (2A, 13, 11) 1760: 1756: 1530: 1113: 863: 751: 739: 735: 301: 297: 259: 255: 190: 166: 206: 21: 1008: 228: 210: 2229:

groups as general individual categories; ten total otherwise), and one of seven classes of simple

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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Correction, there are Lie algebras as part of different maximal subgroups. There's for example G

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is another important link between Lie algebras and the sporadic groups, for example, as are the

73: 52: 1806: 1319: 1289: 1207:. The largest alternating maximal subgroup that exists within sporadic groups is found inside 1148: 1056: 911: 881: 835:) 00:17, 5 March 2023 (UTC) (Note: I misread your original question, instead of subquotients 773:. There are other simple groups that do lie inside the structure of two maximal subgroups of 2544: 2468: 2452: 2447:

Before I change this information, why does the table say 104 dimensions in the first place?

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group) that are maximal subgroups of sporadic groups (see conclusion from prior discussion,

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17 is also the number of primes (inclusive of powers) in the factorization of the order of

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that are shared between these sporadic groups as maximal subgroups, I have thus found only

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It is a middle indexed (order-wise) member in the set of seven Lie groups (when including

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8 9 10 11 12 13 (seven is the middle indexed integer here); with generators (2A, 3, 13),

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secondary sources. I am busy right now, but I will give you one quick secondary source:

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4, bunch of polynomials of a prime power blah blah blah, and 27 sporadic integers. --

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The groups of Lie type can be further divided into 16 families. The full list is at

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It is the seventh largest sporadic group by order, if included as a sporadic group

1607: 1529:. Still, this has been quite a fruitful exercise, so thank you for initiating it @ 1244:(q). There is also one special case when counting a non-strict group of Lie type, 2590: 2540: 2464: 2448: 2398: 2347: 2293: 2269: 2195: 2175: 2103: 2032: 2011: 1953: 1931: 1915: 1876: 1723: 1656: 1596: 1534: 1407: 1220: 1098: 1070:

deeply inside its structure as well. However, in these three groups that share A

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subquotient (e.g. of Lie type) in between. For this a source would be required.

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The Tits group is a subgroup of the Rudvalis group Ru and the Fischer group Fi

1621: 1094:):2):2 of order 72,576,000 and a representation of a permutation on 17 points 458: 316: 79: 1809: 1522:{\displaystyle {\mathsf {G_{a}}}\geq {\mathsf {G_{b}}}\geq {\mathsf {G_{c}}}} 1801: 819:

are not, though, simple on their own. So, we can safely say that there are

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is not overquotient of all sporadics. The Pariahs aren't subquotients of

2501:, of which all sporadics are subquotients ? Let's take the minimal such 2306:

are the series of groups before the higher-realm of two sporadic groups

1976:; which have alternating, sporadic, and other more general types as part 1184:(which is the largest member isomorphic to a Classical Chevalley group, 227:

Perhaps the table of sporadic group orders should be ordered by order?

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of Tits, a source more specific than this article in mathworld.wolfram.

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are deep down at the root level the most basic connecting elements).

1215:. The smallest such group is the (smallest) non-abelian simple group A 432:

of sporadic groups, one group has a maximal subgroup that is the only

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of sporadic groups, only one group has a maximal subgroup of Lie type:

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of group order 2520 is common to three groups as a maximal subgroup:

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double cover of the Tits group as part of their algebraic structure:

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that represents the infinite graded dimensional representation of

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Only one of its kind (not in a family with other groups lacking a

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there is a link from this article to the article "Sporadic Group".

1030:| = 7920, which is one less than 7919 (the 10th prime); the order 769:(2) which contains the simple index 2 somewhat sporadic subgroup 254:

Hiss has Tabelle 2 on page 172, ordered by inventor, invention? –

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any two sporadic group aside from the simple sporadics, and not

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figure is correct, or what qualifying information is missing?-

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prime) 17 is the number of classes of Lie groups if we include

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there are 26 (or 27 if the Tits group 2F4(2)′ is counted also)

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Therefore, following the above enumeration, there are strictly

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is isomorphic to the second smallest simple non-abelian group

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is also the middle index of all 27 sporadic groups, by order.

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as a maximal subgroup, which means that HS has as a subgroup A

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as its largest prime factor dividing its order: 1 2 3 4 5 6

1903: 600:(2)' in the grey zone; and some like in M which contains 2.E 245:

It might, however, be interesting what the initial order is.

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of the 26 sporadics, of which each of them is a proper and

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I don't believe any Lie groups exist inside sporadic groups

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is NOT throughout of Lie type as well. It is, however, an

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T, as a derivative of a full group of Lie type, is only a

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alternating groups as maximal subgroups (and not simply

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as the largest prime number dividing its order (after

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group of an order with prime factor larger than 71? --

2376:) do not contain 7 as a prime factor in their order; 2081: 1470: 1116: 866: 683: 661: 251:

Many authors order by "generation" first, then order.

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That group doesn't exist, so your question is moot.

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representation is over a field that is fourfold 26.

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Lie type derivative and root of an infinite series,

804:, where it is isomorphic to the Classical Chevalley 612:(2), the double cover of the Tits group. Note that G 101:, a collaborative effort to improve the coverage of 2601:you were asking about, and nothing else. Regards, 1521: 1122: 872: 721: 667: 1444:definition of how they are defined — except for F 2173:(the seventh composite on the other hand is 14) 1891:and if you are inclined to help in this regard, 2397:is 21-dimensional, a dimension divisible by 7. 1946:I did a little list of important aspects of T: 1795:Department of Mathematics and Statistics: 106, 1781:"Polytopes Derived from Sporadic Simple Groups" 1779:Hartley, Michael I.; Hulpke, Alexander (2010), 839:sporadic groups, I construed in my mind simply 608:, or Ru where one of its maximal subgroups is F 2519:So, to which infinite systematic family does 2056:turns out to be quite a "meta-expression" of 1849:which these infinite families do not include. 1195:, and all smaller alternating groups. Also, A 8: 2427:Minimal faithful character of the Tits group 1256:sporadic groups. Listed by order, they are: 624:(q); I haven't check for these yet (I know A 158:How many infinite families of simple groups? 2508:It cannot be sporadic, because the monster 2487:, of which all sporadics are subquotients. 2147:is the seventeenth largest sporadic group, 729:. (all groups simple, 2 sporadic) 349:(4) that is a maximal subgroup within Suz. 2362:. Also, the two smallest-sporadic groups ( 1639:You added these a while back in May 2018 ( 1234:finite simple groups that are not sporadic 47: 2258:is the smallest (with a uniquely largest 2000:, and appears as its double cover inside 1800: 1511: 1506: 1505: 1494: 1489: 1488: 1477: 1472: 1471: 1469: 1115: 1015:). This is my addendum, which is notable 865: 698: 697: 685: 684: 682: 660: 361:, last updated 10:47, 4 March 2023 (UTC). 1236:which exist as maximal subgroups inside 1144:of order 20,160 is also common between: 880:Okay, so from a search of finite simple 49: 19: 2414:2607:FEA8:F8E1:F00:95F8:E103:5B90:4504 1577:2607:FEA8:F8E1:F00:95F8:E103:5B90:4504 1562:2607:FEA8:F8E1:F00:95F8:E103:5B90:4504 1512: 1508: 1495: 1491: 1478: 1474: 1023:is in ten dimensions while its order | 708: 705: 702: 699: 689: 686: 2027:There are other important aspects of 1785:Contributions to Discrete Mathematics 1608:Eric W. Weisstein, "Sporadic Group", 821:no simple Lie subquotients in between 545:Worth noting, two groups contain the 319:, to provide a link between the two. 7: 1617:listed among the 26 sporadic groups. 1240:separate sporadic groups; all A or G 248:Sloane, Wolfram etc. order by order. 95:This article is within the scope of 1590:Is the Tits group a sporadic group? 1226:, the only non-pariah Janko group. 675:of the Tits group, so that we have 342:, updated 04:18, 3 March 2023 (UTC) 38:It is of interest to the following 2082:§ No simple subquotient in between 1560:(2), or other "generic" series. -- 315:; that is the very purpose of the 14: 2635:Mid-priority mathematics articles 2557:But there exists, of course, the 1622:Eric W. Weisstein, "Tits Group", 1248:, which would bring the total to 534:contains: (2 × Sz(8)):3 or (2 × B 115:Knowledge:WikiProject Mathematics 2594:was referring to the hypothetic 2283:) -- (note that, when including 1123:{\displaystyle \hookrightarrow } 1062:inside, which in-turn contains A 873:{\displaystyle \hookrightarrow } 800:is alternating and simple like A 286:No simple subquotient in between 118:Template:WikiProject Mathematics 82: 72: 51: 20: 2302:) In the third generation, the 2300:) 03:33, 12 November 2023 (UTC) 2193:, 2 × 3 × 5 × 13 = 17,971,200 1201:projective special linear group 200:Missing an explanatory argument 135:This article has been rated as 2479:Minimal overgroup of sporadics 2247:seven second generation groups 2213:It is one of seven classes of 1164:This brings the total to only 1117: 867: 1: 2422:03:42, 28 February 2024 (UTC) 2407:22:45, 11 November 2023 (UTC) 2204:03:42, 12 November 2023 (UTC) 2184:02:32, 12 November 2023 (UTC) 2112:01:18, 12 November 2023 (UTC) 2041:02:02, 11 November 2023 (UTC) 1940:20:53, 11 November 2023 (UTC) 1924:21:00, 10 November 2023 (UTC) 1885:23:01, 10 November 2023 (UTC) 1732:22:06, 10 November 2023 (UTC) 1665:22:47, 10 November 2023 (UTC) 1585:03:50, 28 February 2024 (UTC) 1570:03:26, 28 February 2024 (UTC) 109:and see a list of open tasks. 2630:C-Class mathematics articles 1765:15:26, 9 November 2023 (UTC) 1602:Very important sources are: 1428:(3) is notably missing (?). 823:any of the sporadic groups. 215:17:03, 27 January 2010 (UTC) 187:list of finite simple groups 179:23:27, 18 October 2008 (UTC) 2217:(when including admissible 1552:, and also eventually of GL 1082:in any form is the group (A 923:, a second generation group 2651: 2483:There exists a group, say 2457:21:45, 30 April 2024 (UTC) 2383:is the first (non-pariah; 1753:infinite systematic family 1678:infinite systematic family 1154:, a first generation group 917:, a first generation group 642:19:30, 23 April 2023 (UTC) 2611:16:43, 16 July 2024 (UTC) 2575:18:41, 14 July 2024 (UTC) 2549:18:04, 13 July 2024 (UTC) 2473:17:13, 13 July 2024 (UTC) 2328:(that itself fits inside 1706:, in the infinite family 1676:it does NOT belong to an 1543:05:36, 6 March 2023 (UTC) 1107:03:04, 5 March 2023 (UTC) 853:05:56, 6 March 2023 (UTC) 744:16:54, 4 March 2023 (UTC) 359:05:52, 4 March 2023 (UTC) 340:15:13, 3 March 2023 (UTC) 306:14:14, 2 March 2023 (UTC) 280:15:09, 3 March 2023 (UTC) 264:14:07, 2 March 2023 (UTC) 194:13:19, 18 June 2009 (UTC) 134: 67: 46: 2534:16:43, 26 May 2024 (UTC) 2241:is not arbitrary in the 2140:Looking more carefully, 1956:, or other of the like), 1680:of finite simple groups. 1203:PSL(3,2), found inside M 141:project's priority scale 2245:, since there are also 1802:10.11575/cdm.v5i2.61945 1747:of Ree groups of type F 1714:of Lie type for n : --> 237:04:17, 1 May 2010 (UTC) 98:WikiProject Mathematics 2155:, which embeds inside 1908: 1523: 1124: 874: 723: 669: 28:This article is rated 1907: 1793:University of Calgary 1524: 1372:, of order 17,971,200 1132:alternating group on 1125: 895:alternating group on 875: 724: 670: 2151:taking into account 1613:, the Tits group is 1468: 1452:) which stems from F 1114: 864: 681: 659: 559:(pariah) contains: F 121:mathematics articles 1972:It is one of seven 1419:order 5,859,375,000 1252:such groups inside 1219:that exists within 1130:Update: The simple 2497:Is there a simple 2346:-expansion of the 1909: 1791:(2), Alberta, CA: 1519: 1120: 882:alternating groups 870: 719: 665: 436:group of Lie type: 90:Mathematics portal 34:content assessment 2119:It embeds inside 1993:It embeds inside 1899:And look at this: 1403:order 251,596,800 1387:order 239,500,800 668:{\displaystyle X} 382:second generation 169:comment added by 155: 154: 151: 150: 147: 146: 2642: 2593: 2585: 2301: 2206: 2186: 2114: 2050:More precisely, 1985:quasithin groups 1850: 1833: 1804: 1746: 1713: 1704:exceptional case 1600: 1528: 1526: 1525: 1520: 1518: 1517: 1516: 1515: 1501: 1500: 1499: 1498: 1484: 1483: 1482: 1481: 1129: 1127: 1126: 1121: 1044: 1042: 1035: 1029: 879: 877: 876: 871: 755: 728: 726: 725: 720: 712: 711: 693: 692: 674: 672: 671: 666: 654: 480:contains: (2 × F 470:contains: (3 × G 430:third generation 371:first generation 328:quasithin groups 314: 181: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 2650: 2649: 2645: 2644: 2643: 2641: 2640: 2639: 2620: 2619: 2588: 2579: 2481: 2442: 2429: 2395: 2388: 2381: 2374: 2367: 2360: 2340: 2333: 2326: 2315: 2292: 2281: 2255: 2194: 2174: 2160: 2145: 2135: 2124: 2102: 1998: 1910: 1874: 1870: 1848: 1778: 1750: 1744: 1740: 1721: 1711: 1707: 1696:degenerate case 1594: 1592: 1559: 1555: 1507: 1490: 1473: 1466: 1465: 1455: 1447: 1435: 1427: 1416: 1400: 1385: 1365: 1357: 1346: 1337: 1323: 1313: 1293: 1282: 1273: 1265: 1243: 1224: 1218: 1214: 1206: 1198: 1194: 1191:(2)) is found A 1190: 1183: 1179: 1175: 1152: 1142: 1112: 1111: 1093: 1089: 1085: 1081: 1077: 1073: 1069: 1065: 1060: 1041: 1037: 1033: 1031: 1028: 1024: 1022: 1002: 998: 994: 990: 986: 979: 975: 968: 964: 957: 950: 946: 938: 915: 906: 862: 861: 818: 814: 810: 803: 799: 795: 788: 784: 780: 768: 764: 749: 734:Is there one? – 679: 678: 657: 656: 648: 631: 627: 623: 615: 611: 607: 603: 599: 595: 591: 580: 576: 566: 562: 537: 523: 497: 493: 483: 473: 454: 446: 420: 416: 411: 399: 348: 324: 312: 288: 225: 202: 164: 160: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 2648: 2646: 2638: 2637: 2632: 2622: 2621: 2618: 2617: 2616: 2615: 2614: 2613: 2559:direct product 2552: 2551: 2524: 2480: 2477: 2476: 2475: 2445: 2444: 2440: 2437: 2428: 2425: 2393: 2386: 2379: 2372: 2365: 2358: 2338: 2331: 2324: 2313: 2304:Fischer groups 2279: 2253: 2243:classification 2234: 2233: 2210: 2209: 2208: 2207: 2187: 2158: 2143: 2138: 2133: 2128: 2127: 2122: 2116: 2115: 2098: 2097: 2086: 2069: 2048: 2046: 2044: 2043: 2025: 2024: 2023: 2008: 1996: 1990: 1989: 1988: 1987: 1978: 1977: 1969: 1968: 1967: 1966: 1960: 1959: 1958: 1957: 1943: 1942: 1927: 1926: 1912: 1911: 1902: 1900: 1896: 1895: 1872: 1871:, as well as A 1868: 1859: 1858: 1857: 1856: 1855: 1854: 1837: 1836: 1835: 1834: 1773: 1772: 1748: 1742: 1737: 1736: 1735: 1734: 1719: 1709: 1682: 1681: 1670: 1669: 1668: 1667: 1633: 1629: 1628: 1618: 1601: 1591: 1588: 1573: 1572: 1557: 1553: 1514: 1510: 1504: 1497: 1493: 1487: 1480: 1476: 1453: 1445: 1433: 1425: 1421: 1420: 1414: 1404: 1398: 1388: 1383: 1373: 1363: 1355: 1350: 1344: 1335: 1321: 1316: 1311: 1291: 1286: 1280: 1271: 1263: 1241: 1222: 1216: 1212: 1204: 1196: 1192: 1188: 1181: 1177: 1173: 1162: 1161: 1155: 1150: 1140: 1119: 1091: 1087: 1083: 1079: 1075: 1071: 1067: 1063: 1058: 1039: 1026: 1020: 1000: 996: 992: 991:— in between A 988: 984: 977: 973: 966: 962: 955: 948: 944: 936: 933: 932: 931: 930: 924: 918: 913: 904: 869: 859: 858: 857: 856: 816: 812: 808: 801: 797: 793: 786: 782: 778: 766: 762: 732: 731: 730: 718: 715: 710: 707: 704: 701: 696: 691: 688: 664: 629: 625: 621: 613: 609: 605: 601: 597: 593: 589: 585: 584: 583: 582: 578: 574: 568: 564: 560: 551: 550: 542: 541: 540: 539: 535: 526: 521: 510: 509: 501: 500: 499: 498: 495: 491: 485: 481: 475: 471: 462: 452: 444: 438: 437: 425: 424: 423: 422: 418: 414: 409: 402: 397: 386: 385: 378: 375:Mathieu groups 363: 362: 346: 343: 322: 291: 287: 284: 283: 282: 267: 266: 252: 249: 246: 243: 224: 221: 219: 201: 198: 197: 196: 159: 156: 153: 152: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 2647: 2636: 2633: 2631: 2628: 2627: 2625: 2612: 2608: 2604: 2600: 2597: 2592: 2583: 2578: 2577: 2576: 2572: 2568: 2564: 2560: 2556: 2555: 2554: 2553: 2550: 2546: 2542: 2538: 2537: 2536: 2535: 2531: 2527: 2522: 2517: 2515: 2511: 2506: 2504: 2500: 2495: 2493: 2488: 2486: 2478: 2474: 2470: 2466: 2461: 2460: 2459: 2458: 2454: 2450: 2438: 2434: 2433: 2432: 2426: 2424: 2423: 2419: 2415: 2409: 2408: 2404: 2400: 2396: 2389: 2382: 2375: 2368: 2361: 2354: 2352: 2351: 2345: 2341: 2334: 2327: 2320: 2316: 2309: 2305: 2299: 2295: 2290: 2286: 2282: 2276:, and before 2275: 2271: 2267: 2266: 2261: 2257: 2256: 2248: 2244: 2240: 2239: 2232: 2228: 2227: 2222: 2221: 2216: 2212: 2211: 2205: 2201: 2197: 2192: 2188: 2185: 2181: 2177: 2172: 2168: 2164: 2163: 2161: 2154: 2150: 2146: 2139: 2136: 2130: 2129: 2125: 2118: 2117: 2113: 2109: 2105: 2100: 2099: 2095: 2091: 2087: 2084: 2083: 2078: 2074: 2070: 2067: 2066: 2065: 2063: 2059: 2055: 2051: 2047: 2042: 2038: 2034: 2030: 2026: 2021: 2017: 2013: 2009: 2007: 2003: 1999: 1992: 1991: 1986: 1982: 1981: 1980: 1979: 1975: 1971: 1970: 1964: 1963: 1962: 1961: 1955: 1951: 1950: 1948: 1947: 1945: 1944: 1941: 1937: 1933: 1929: 1928: 1925: 1921: 1917: 1914: 1913: 1906: 1901: 1898: 1897: 1894: 1890: 1886: 1882: 1878: 1866: 1861: 1860: 1852: 1851: 1843: 1842: 1841: 1840: 1839: 1838: 1832: 1829: 1825: 1822: 1818: 1815: 1811: 1808: 1803: 1798: 1794: 1790: 1786: 1782: 1777: 1776: 1775: 1774: 1769: 1768: 1767: 1766: 1762: 1758: 1754: 1733: 1729: 1725: 1717: 1716: 1705: 1701: 1697: 1690: 1686: 1685: 1684: 1683: 1679: 1675: 1674: 1673: 1666: 1662: 1658: 1654: 1650: 1646: 1642: 1638: 1637: 1636: 1635: 1634: 1626: 1625: 1619: 1616: 1612: 1611: 1605: 1604: 1603: 1598: 1589: 1587: 1586: 1582: 1578: 1571: 1567: 1563: 1551: 1547: 1546: 1545: 1544: 1540: 1536: 1532: 1502: 1485: 1463: 1459: 1451: 1441: 1439: 1438:cyclic groups 1429: 1418: 1410: 1409: 1405: 1402: 1394: 1393: 1389: 1386: 1379: 1378: 1374: 1371: 1367: 1359: 1358: 1351: 1348: 1343: 1338: 1331: 1330: 1325: 1324: 1317: 1314: 1307: 1306: 1301: 1300: 1295: 1294: 1287: 1284: 1279: 1274: 1267: 1266: 1259: 1258: 1257: 1255: 1251: 1247: 1239: 1235: 1232: 1227: 1225: 1210: 1202: 1187: 1171: 1167: 1159: 1156: 1153: 1147: 1146: 1145: 1143: 1137: 1135: 1109: 1108: 1104: 1100: 1096: 1061: 1054: 1049: 1048: 1018: 1014: 1013:Leech lattice 1010: 1006: 1005:Euler totient 983: 972: 961: 954: 943: 928: 925: 922: 919: 916: 910: 909: 907: 900: 898: 892: 891: 890: 887: 883: 854: 850: 846: 842: 838: 834: 830: 826: 822: 807: 792: 776: 772: 760: 753: 748:My pleasure! 747: 746: 745: 741: 737: 733: 716: 713: 694: 677: 676: 662: 652: 647: 646: 645: 643: 639: 635: 620: 572: 569: 558: 555: 554: 553: 552: 548: 547:almost simple 544: 543: 533: 530: 529: 527: 525: 517: 516: 512: 511: 507: 503: 502: 490:contains: 2.E 489: 486: 479: 476: 469: 466: 465: 463: 460: 456: 448: 447: 440: 439: 435: 431: 427: 426: 413:contains : (A 412: 406: 405: 403: 401: 393: 392: 388: 387: 383: 379: 376: 372: 368: 367: 366: 360: 356: 352: 344: 341: 337: 333: 329: 325: 318: 310: 309: 308: 307: 303: 299: 295: 285: 281: 277: 273: 269: 268: 265: 261: 257: 253: 250: 247: 244: 241: 240: 239: 238: 234: 230: 222: 220: 217: 216: 212: 208: 199: 195: 192: 188: 184: 183: 182: 180: 176: 172: 171:128.220.30.95 168: 157: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 2598: 2595: 2520: 2518: 2513: 2509: 2507: 2502: 2498: 2496: 2491: 2489: 2484: 2482: 2446: 2430: 2410: 2391: 2384: 2377: 2370: 2363: 2356: 2349: 2348: 2343: 2336: 2329: 2322: 2318: 2288: 2284: 2277: 2273: 2263: 2259: 2251: 2246: 2236: 2235: 2224: 2218: 2190: 2170: 2166: 2156: 2152: 2148: 2141: 2131: 2120: 2093: 2089: 2088:It contains 2080: 2076: 2072: 2061: 2057: 2053: 2052: 2049: 2045: 2028: 2019: 2015: 2005: 2001: 1994: 1892: 1888: 1864: 1846: 1845: 1788: 1784: 1752: 1738: 1700:special case 1693: 1692: 1688: 1677: 1671: 1630: 1623: 1614: 1609: 1593: 1574: 1549: 1461: 1457: 1449: 1442: 1430: 1422: 1412: 1406: 1396: 1390: 1381: 1375: 1369: 1361: 1352: 1349:order 20,160 1341: 1340: 1333: 1327: 1318: 1309: 1303: 1297: 1288: 1277: 1276: 1269: 1260: 1253: 1249: 1245: 1237: 1233: 1230: 1228: 1211:, which is A 1185: 1169: 1165: 1163: 1138: 1133: 1131: 1110: 1050: 1046: 987:(2); where A 981: 970: 959: 952: 941: 934: 902: 896: 894: 885: 860: 840: 836: 824: 820: 805: 790: 774: 770: 758: 618: 586: 546: 519: 513: 505: 461:T on its own 450: 441: 433: 429: 395: 389: 381: 370: 364: 289: 226: 218: 203: 161: 137:Mid-priority 136: 96: 62:Mid‑priority 40:WikiProjects 2249:, of which 2215:thin groups 2012:group order 1448:(2)′ (also 1411:: contains 1395:: contains 1380:: contains 1315:order 2,520 1268:: contains 947:(q), only A 893:The simple 815:.2 and 5:4A 781:.2 and 5:4A 596:(5), with F 573:contains: S 504:Within the 428:Within the 380:Within the 369:Within the 223:Table order 165:—Preceding 112:Mathematics 103:mathematics 59:Mathematics 2624:Categories 2582:Nomen4Omen 2567:Nomen4Omen 2565:subgroup. 2526:Nomen4Omen 2353:-invariant 2077:non-strict 1831:1320.51021 1757:Nomen4Omen 1556:(2), or Sp 1550:eventually 1531:Nomen4Omen 1462:in-between 1332:: contain 1308:: contain 1160:, a pariah 976:(9), and A 929:, a pariah 837:in-between 752:Nomen4Omen 736:Nomen4Omen 518:contains: 459:Tits group 449:contains: 394:contains: 317:Tits group 298:Nomen4Omen 256:Nomen4Omen 191:Algebraist 2523:belong ? 2494:simple ? 2062:seventeen 1810:1715-0868 1694:T is the 1624:MathWorld 1610:MathWorld 1360:contains 1009:moonshine 785:, where A 592:(4) and G 434:nonstrict 207:Meerassel 2603:JoergenB 2231:N-groups 2162:, where 1974:N-groups 1824:40845205 1285:order 60 1170:subroups 1036:| = 100| 1011:and the 229:User4096 167:unsigned 2321:inside 2167:seventh 1954:BN-pair 1893:coolio. 1817:2791293 1464:, i.e. 1458:between 1136:letters 899:letters 841:between 563:(2) = F 506:pariahs 139:on the 30:C-class 2596:simple 2591:Radlrb 2563:normal 2541:Radlrb 2465:Radlrb 2449:Blobs2 2399:Radlrb 2335:, and 2317:; so, 2294:Radlrb 2196:Radlrb 2176:Radlrb 2104:Radlrb 2033:Radlrb 2004:, and 1932:Radlrb 1916:Radlrb 1877:Radlrb 1724:Radlrb 1657:Radlrb 1597:Radlrb 1535:Radlrb 1099:Radlrb 1055:holds 965:(5), A 958:(4) ≃ 845:Radlrb 829:Radlrb 811:(9). A 796:(9). A 651:Radlrb 634:Radlrb 567:(2)'.2 538:(8)):3 484:(2)):2 474:(3)):2 457:; the 421:(4)):2 351:Radlrb 332:Radlrb 294:simple 272:Radlrb 36:scale. 2165:(the 2075:as a 2060:(and 2058:seven 1821:S2CID 1702:, or 1698:, or 1368:, or 1250:seven 1176:and A 1134:eight 999:and A 897:seven 825:Ta-da 628:and A 604:(2):S 494:(2):S 2607:talk 2571:talk 2545:talk 2530:talk 2469:talk 2453:talk 2436:too. 2418:talk 2403:talk 2369:and 2310:and 2298:talk 2223:and 2200:talk 2180:talk 2149:sans 2108:talk 2037:talk 2010:Its 1936:talk 1920:talk 1881:talk 1807:ISSN 1761:talk 1745:(2)′ 1728:talk 1712:(2)′ 1689:root 1661:talk 1581:talk 1566:talk 1539:talk 1366:(2)' 1326:and 1302:and 1283:(4) 1238:nine 1103:talk 1086:× (A 1017:imho 849:talk 833:talk 740:talk 638:talk 455:(2)' 355:talk 336:talk 302:talk 276:talk 260:talk 233:talk 211:talk 175:talk 2490:Is 2289:Suz 2265:Suz 2226:PSU 2220:PSL 1828:Zbl 1797:doi 1620:In 1615:not 1606:In 1417:(5) 1401:(4) 1392:Suz 1347:(2) 1305:O'N 1299:Suz 1254:ten 1231:six 1166:two 1090:× A 1047:\o/ 995:, A 927:O'N 921:Suz 886:one 777:: A 581:(2) 577:× F 524:(5) 417:× G 400:(4) 391:Suz 373:of 131:Mid 2626:: 2609:) 2573:) 2547:) 2532:) 2516:. 2505:. 2471:) 2455:) 2441:22 2420:) 2405:) 2394:22 2380:22 2373:12 2366:11 2339:24 2337:Fi 2332:23 2330:Fi 2325:22 2323:Fi 2287:, 2280:22 2278:Fi 2270:13 2202:) 2182:) 2159:22 2157:Fi 2144:22 2142:Fi 2134:22 2132:Fi 2123:22 2121:Fi 2110:) 2090:13 2039:) 2022:). 2016:17 2002:Ru 1997:22 1995:Fi 1938:) 1922:) 1889:or 1883:) 1869:11 1826:, 1819:, 1814:MR 1812:, 1805:, 1787:, 1783:, 1763:) 1730:) 1720:22 1715:0; 1663:) 1651:, 1647:, 1643:, 1583:) 1568:) 1558:2n 1541:) 1503:≥ 1486:≥ 1408:Ly 1384:12 1377:HN 1356:22 1354:Fi 1339:≃ 1329:Ru 1322:23 1296:, 1292:22 1275:≃ 1213:12 1209:HN 1205:22 1158:Ru 1151:23 1118:↪ 1105:) 1097:. 1076:22 1074:(M 1059:22 1053:HS 1040:22 1034:HS 1027:11 1021:11 980:≃ 969:≃ 951:≃ 937:22 914:22 901:, 868:↪ 855:). 851:) 789:≃ 742:) 714:≥ 695:≥ 644:. 640:) 557:Ru 532:Ru 515:Ly 468:Th 445:22 443:Fi 408:Co 357:) 338:) 330:. 304:) 278:) 262:) 235:) 213:) 189:. 177:) 2605:( 2599:L 2589:@ 2584:: 2580:@ 2569:( 2543:( 2528:( 2521:L 2514:M 2510:M 2503:L 2499:L 2492:L 2485:L 2467:( 2451:( 2416:( 2401:( 2392:M 2387:1 2385:J 2378:M 2371:M 2364:M 2359:1 2357:F 2350:j 2344:q 2319:T 2314:1 2312:F 2308:B 2296:( 2285:T 2274:T 2260:p 2254:2 2252:J 2238:7 2198:( 2191:T 2178:( 2171:T 2153:T 2106:( 2094:7 2085:) 2073:T 2054:T 2035:( 2029:T 2020:T 2006:B 1934:( 1918:( 1879:( 1873:7 1853:" 1799:: 1789:5 1759:( 1749:4 1743:4 1741:F 1726:( 1710:4 1708:F 1659:( 1653:4 1649:3 1645:2 1641:1 1599:: 1595:@ 1579:( 1564:( 1554:n 1537:( 1513:c 1509:G 1496:b 1492:G 1479:a 1475:G 1454:4 1450:T 1446:4 1434:3 1426:2 1424:G 1415:2 1413:G 1399:2 1397:G 1382:A 1370:T 1364:4 1362:F 1345:3 1342:A 1336:8 1334:A 1320:M 1312:7 1310:A 1290:M 1281:1 1278:A 1272:5 1270:A 1264:2 1262:J 1246:T 1242:2 1223:2 1221:J 1217:5 1197:8 1193:7 1189:3 1186:A 1182:8 1178:8 1174:7 1149:M 1141:8 1139:A 1101:( 1092:5 1088:5 1084:7 1080:7 1072:7 1068:7 1064:7 1057:M 1043:| 1038:M 1032:| 1025:M 1001:8 997:6 993:5 989:7 985:3 982:A 978:8 974:1 971:A 967:6 963:1 960:A 956:1 953:A 949:5 945:n 942:A 935:M 912:M 905:7 903:A 847:( 831:( 817:4 813:6 809:1 806:A 802:6 798:4 794:1 791:A 787:6 783:4 779:6 775:T 771:T 767:4 763:4 759:T 754:: 750:@ 738:( 717:X 709:s 706:t 703:i 700:T 690:u 687:R 663:X 653:: 649:@ 636:( 630:8 626:5 622:n 619:A 614:2 610:4 606:3 602:6 598:4 594:2 590:2 588:G 579:4 575:4 571:B 565:4 561:4 536:2 522:2 520:G 496:3 492:6 488:M 482:4 478:B 472:2 453:4 451:F 419:2 415:4 410:1 398:2 396:G 353:( 347:2 334:( 323:8 321:E 300:( 274:( 258:( 231:( 209:( 173:( 143:. 42::

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