How much of the ATLAS of finite groups is independently checked and/or computer verified? In a recent talk Finite groups, yesterday and today Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the CFSG and said so was ok, but a proof that relied on the ATLAS was not so ok, because the content has not been completely independently verified, and has as its basis old computer and other calculations that have only been done once.
Given that (numerous?) little errors have been found over time, it would be good to have a definitive record as to which bits have either been calculated or proved elsewhere, or formally verified in the case of the computer calculations, and if so where and by whom (with code for the latter case). Note that merely being able to do some calculations in GAP is not quite enough, since, as the documentation says

Part of the constructions have been documented in the literature on almost simple groups, or the results have been used in such publications, see for example the references in [CCNPW85] and [BN95].

where CCNPW85 is the ATLAS and BN95 is Breuer and Norton's Improvements to the Atlas (in an appendix of Atlas of Brauer Characters).

EDIT Since it may not have been clear, I was after statements modeled on the following:

*

*"All results about classical groups of Lie type are well-known and documented elsewhere"

*"All results about conjugacy classes of the sporadic groups except Janko 4 [say] are calculated afresh and contained in X computer package"

*"The results on [blah] about [some group] are only contained in the ATLAS, and no papers or independently written software have reproved/recalculated them"

If it's easier to specify what is only in the ATLAS then that would be good, since clearly a lot of the classical material would be known and calculated long before.

EDIT May 2017 In a recent talk (50 years ago: a great time for number theory — the first few minutes only before the main talk) Serre mentions his comments discussed here, the fact people got worked up about it, and the paper Breuer, Malle, and O'Brien - Reliability and reproducibility of Atlas information in Farrokh Shirjian's answer, which he feels addresses his complaints.
 A: At the recent 'Conway and the Atlases' conference in Princeton it was remarked that computers can easily check everything in the Atlas up to about J4. With regards to references, it is worth noting that whilst the bibliography in the original Atlas is now thirty years old, a very comprehensive update was given in the Atlas of Brauer Characters by Jansen, Lux, Parker and Wilson, published ten years later. It's also worth noting that whilst oldest edition of the Atlas does indeed contain numerous errors, more recent editions contain an 'addenda and corrigenda' section that corrects many of them, the most serious being the omission of some columns from one character table. 
A: I must disagree with Serre (or your way to interpret his words) here. There are no papers that rely on "the" Atlas, they inevitably only use a tiny part, e.g. the character table of a particular group, information about the structure of a particular subgroup in a particular group, etc. As such, it's totally OK for such a paper; indeed there are probably few places in Atlas which were not checked independently, but these are probably almost never used.
Atlas is an atlas - like all the maps, there are few tiny errors here and there, but still it is more reliable than a theorem (CFSG) with a proof so long that one can probably count people who understand it in full detail on fingers of one hand...
And saying that one must not rely on Atlas for information that is well-known, e.g. maximal subgroups of "smallish" sporadic simple groups, something that was independently checked in many sources, is just silly. As well, many computations of character tables were re-done independently, by people working on GAP and Cayley (Magma), and many tables in the Atlas for non-sporadic groups follow from general theory (e.g. character tables of alternating groups aren't some kind of thing one cannot find elsewhere).
A: Unlike Dima, I am inclined to agree with Serre on this point. Although most of the facts recorded in the ATLAS have been proved elsewhere or, in the case of all of the character tables except for those of the very large groups like the Monster, can be easily recomputed in GAP or Magma using standard algorithms for finite groups, it can be very difficult in some cases to track down alternative proofs.
I have recently completed a book (co-authored with John Bray and Colva Roney-Dougal) calculating complete lists of maximal subgroups of almost simple classical groups in dimensions up to $12$, and we were confronted with this problem. Although we cited the ATLAS many times, we tried hard to provide alternative citations or, for facts that could be easily checked by computer, we provided code to do this. In fact nearly all of the facts we required were either about maximal subgroups of groups in the ATLAS or involved entries in character tables. For the sporadic groups there were virtually always alternative papers to cite, which were generally also cited in the ATLAS.
We had more difficulties with things like maximal subgroups of almost simple extensions of some of the more complicated classical groups in the ATLAS, like $U_4(3)$. For these we could not always find alternative sources that gave precise enough information, and we were told informally that some of the information had been originally  calculated by unidentified research students or PostDocs. So we tried hard to re-prove these facts.
Having said that, we found remarkably few errors in the ATLAS. I think there might have been one or two very small and minor inaccuracies in some of the structure descriptions, which we reported to the authors, and I think they might have known about them already. I see that there is an "ATLAS 30 Years On" conference coming up in Princeton in November 2015, so perhaps this will lead to more discusssion of these questions.
I should also reinforce the point made by David Roberts that one needs to be very cautious when using Computer Algebra Systems, such as GAP and Magma, to verify facts contained in sources like the ATLAS, because it is possible that the code used is itself relying on these sources. For example $\mathtt{ MaximalSubgroups}$ in Magma will generally look up the maximal subgroups of the group's composition factors in a database, which will have been constructed using the ATLAS. However, a default use of $\mathtt{CharacterTable}$ on a finite group will use a general purpose algorithm (such as Dixon-Schneider), which does not rely on properties of specific (simple) groups. 
A: The Atlas was a work of scholarship, not research.  Our aim in those days was to collect information together for convenience, and the large character tables, and all the other data, was all proved.  The fact that there were so many errors is down to human error - both in our sources and our own work.
Nevertheless it would be possible to re-compute much of it now fairly easily, and in particular computing the character tables of explicitly given matrices or permutations would be less heroic now than 30 years ago.  The result would be that there exists a group with the proven properties.  If anyone wants to have a go at that, I'd love to help.
The "uniqueness" part is more difficult.  There is much published work on properties of simple groups, and the definition of the group studied would need to be somehow be connected to the (re)computed data.  This would require further work.  For example one would need to show for the first Conway group that the 24x24 matrices used to define the group fix an even unimodular lattice in 24 dimensions with minimum norm 4 and that there is an involution centralizer of the form 2.1+8.O8+(2) and so on.
There is no doubt that the project is feasible, and I had a short discussion with Serre on the subject.  Let's hope someone cares enough to actually do it!
A: It may also worth to look at the paper

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*T. Breuer, G. Malle, and E. A. O'Brien, Reliability and reproducibility of Atlas information, Contemporary Mathematics 694 (2017) pp 21–31, doi:10.1090/conm/694/13960, arXiv:1603.08650
in which is discussed the reliability and reproducibility of much of the information contained in the Atlas of Finite Groups.

(Added by David Roberts) There are the follow-up papers

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*Thomas Breuer, Constructing the ordinary character tables of some Atlas groups using character theoretic methods, https://arxiv.org/abs/1604.00754,

and

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*Thomas Breuer, Kay Magaard, Robert Wilson, Verification of the ordinary character table of the Baby Monster, https://arxiv.org/abs/1902.07758,

which bring the verification project to the point of double checking all the ATLAS character tables except that of the Monster, $\mathbb{M}$.
Finally, I found the slides Verifying the Character Table of the Monster, from a presentation on 12 December 2020, giving the work of Breuer, Magaard and Wilson on independently constructing the Monster's character table, modulo checking the tables for the following normaliser subgroups of $\mathbb{M}$:

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*$N(2B)\simeq 2^{1+24}_+.\mathrm{Co}_1$, due to Norton;

*$N(3B) \simeq 3^{1+12}_+.2.\mathrm{Suz}.2$, due to Barraclough and Wilson in The Character Table of a Maximal Subgroup of the Monster; and

*$N(5B)\simeq 5^{1+6}_+.4.J_2.2$.

