# Reference request: a list of (small) finite simple groups

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

• Thanks! How (or where) did you find these? Mar 31 '20 at 20:45
• 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. Apr 1 '20 at 7:44