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.


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

  • $\begingroup$ 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


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


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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