A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low degrees, and I'd like to determine where to search for preexisting computations.

Specifically, I am interested in:

- Finite groups such as $D_{2n}$; $A_n$ and $S_n$ for $n\le 5$; binary dihedral/tetrahedral/etc. groups. In general, small and/or well-studied groups.
- The ring structures for their group cohomology with $\mathbb Z$ and $\mathbb Z/2$ coefficients, at least in degrees up to about 6.

I don't mind computing these myself, but if they've already been computed it would be nice to know where to look, and these are the kinds of groups whose cohomology rings have presumably already been computed. But when I search for such computations, I can usually find cohomology groups, but not the ring structure. I suspect I'm looking in the wrong places.

So, **what are some good starting places to search for preexisting computations of the ring structure on group
cohomology?**

Examples of what I might hope for:

- It would be wonderful if there were a database with this information, but this is a lot to hope for.
- I've had some luck finding computations in papers which use group cohomology on the way to some other result (e.g. computing bordism groups of $BG$), but there are surely plenty of applications I'm unaware of, so if you know of applications that require these kinds of computations as input, I'd be interested in hearing about them.

"On products in the cohomology of the Dihedral groups". $\endgroup$ – Chris Gerig Nov 21 '19 at 20:36