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I'm planning to make a video that shows color coded group tables for all 2328 groups of order 128 -- at 128$\times$128 pixels at 24 frames a second I think I get 97 seconds of video. Is there some logical order that the all these groups can be placed in that would make the most visual sense?

It's done! Here: http://www.youtube.com/watch?v=25Qzu-KuVs4

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You might experiment first with, say, all the groups of order 16. Look at the order in which they appear in the various sources that list them, and see whether any make visual sense. –  Gerry Myerson Feb 3 '12 at 11:14
Nice question, I don't have an answer for! But I would like to see such a video after you have made one. Please share the link here with us after you have made the video! –  user21436 Feb 3 '12 at 11:14
It would, perhaps, make sense to put the abelian ones together, as a group is abelian if and only if it has symmetric group table. However, only a small fraction of groups of order $128$ are abelian! –  user1729 Feb 3 '12 at 11:20
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2 Answers

I would be inclined to follow the ordering in the GAP/Magma libraries, which sorts first by number of generators. So you get the cyclic group (tall and thin) first and the elementary abelian group (short and wide) last. Of course the huge majority of the groups have 3,4 or 5 generators.

Groups with the same generator number are sorted in ascending order of exponent $p$-class (which is similar to but not quite the same as nilpotency class). But again you get huge numbers of group with, for example, 4 generators and class 2, which are all very similar to each other. So perhaps the video should go slowly at the beginning and end, when group properties are changing more rapidly, and speed up during the repetitive middle sections!

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Something like this?

Quandles of Order 5

(This is a movie of the Cayley tables of quandles of order five. I could not upload the order seven file; I guess it was too big.) I generated these movies in row-major lexicographic order. You could do the same or, alternatively, you could do those groups in the order they occur in the GAP/Magma libraries of small groups.

But, for groups of order $128$ there might be a partial order that would be visually interesting: sort by the size of the centre. This would put the abelian ones together. For groups with the same size centres, you could recursively sort according to the same criterion applied to their central factor groups. Since your groups are $2$-groups, this will eventually terminate, and you more-or-less work your way along the upper central series. (When you get down to abelian groups, you could break ties by looking at the vectors of sizes of cyclic factors.) Alternatively, you could go down instead of up.

EDIT: Out of curiosity, I tried doing the latter (going down the lower central series) for the groups of orders 16 and 32. (Let's see if I can upload the more interesting order 32 movie ... nope. Let's try the order 16 ... okay.) It is perhaps not as exciting as I had first imagined, though you might have better luck with the upper central series (or lower $2$-central series). I didn't try playing with the colours or frame rate, which might be more usefully chosen to reflect the structure.

Groups of order 16

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Nice! I chose $|G|=128$ because I didn't want there to be too many groups, but I wanted it to be large enough that I could show interesting behaviour. In the case of your second animation, (and I'm not sure how I'm going to color elements), would it be more useful to have the elements ordered by hue on the top and side? –  deoxygerbe Feb 4 '12 at 3:07
I think so. You might even experiment with grey scale. Much more important, however, is ordering the group elements (for each group, that is) in a way that is compatible with the subgroup chain. E.g., sort elements of the smallest subgroup before those in the next, and so on. I didn't realise I hadn't done that here until I was driving home for the weekend. It's an interesting project you've set yourself. Please post a link to your work when you're done! –  James Feb 6 '12 at 20:26
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