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I am currently in the midst of a project in which it would be useful to have a list of all (small) simple groups as a means to check calculations, not waste time, verify conjectures for small examples, etc.

I found this list which enumerates all groups of order $\leq 100$. This tells me that something like this is technically possible, and likely already exists, but I've not been able to find it.

Edit: Perhaps I should have mentioned this: I do not want something like the wikipedia page which has a table of the different types of simple groups. I would like something similar to the first link, which lists all groups with order $x$, then all groups of order $x+1$, etc.


I'm not sure how much more specificity I can add, but I'd be happy to answer any questions if I'm unclear.

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Here is a list of orders of nonabelian simple groups up to 10000. Of course, in addition, there is a abelian simple group of order each prime. You can already see from this short list that the most frequently occurring type of group is ${\rm PSL}(2,q)$ for prime powers $q$.

60: $A_5 \cong {\rm PSL}_2(4) \cong {\rm PSL}_2(5)$.

168: ${\rm PSL}_2(7) \cong {\rm PSL}_3(2)$.

360: $A_6 \cong {\rm PSL}_2(9)$.

504: ${\rm PSL}_2(8)$.

660: ${\rm PSL}_2(11)$.

1092: ${\rm PSL}_2(13)$.

2448: ${\rm PSL}_2(17)$.

2520: $A_7$.

3420: ${\rm PSL}_2(19)$.

4080: ${\rm PSL}_2(16)$.

5616: ${\rm PSL}_3(3)$.

6048: ${\rm PSU}_3(3)$.

6072: ${\rm PSL}_2(23)$.

7800: ${\rm PSL}_2(25)$.

7920: $M_{11}$.

9828: ${\rm PSL}_2(27)$.

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  • $\begingroup$ Thanks! How (or where) did you find these? $\endgroup$ Mar 31 '20 at 20:45
  • $\begingroup$ I got them from the ATLAS of Finite Groups, but it is straightforward to calculate such lists using the classification of finite simple groups, and verret gave a link to a much longer such list. $\endgroup$
    – Derek Holt
    Apr 1 '20 at 7:44
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http://www.madore.org/~david/math/simplegroups.html

Here is a list (in the nonabelian case), up to ten billion.

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Use GAP! It has detailed knowledge of many small groups out of the box, plays nicely with Sage, and lets you check calculations, etc., by writing programs.

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