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What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx 1{,}000{,}000$ elements.

I'll start the ball rolling with the five platonic solids. (Unfortunately not large.)

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closed as too localized by Grumpy Parsnip, Pete L. Clark, Ilya, J. M., Asaf Karagila Oct 20 '11 at 11:31

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6 775 235 700 (en.wikipedia.org/wiki/World_population) is a bit bigger then 2^20, but still it is a finite set of such cardinality where all of us are elements. –  Ilya Oct 19 '11 at 15:12
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@Gortaur: "in mathematics." –  Qiaochu Yuan Oct 19 '11 at 15:13
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I can't wait until Asaf sees this question... –  Willie Wong Oct 19 '11 at 15:19
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I don't understand the motivation behind this question. –  Grumpy Parsnip Oct 19 '11 at 15:43
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This is not actually a question in the sense of this site. It is a call for a discussion, or referendum, on people's tastes and preferences. I have voted to close. –  Pete L. Clark Oct 19 '11 at 16:58

2 Answers 2

The sporadic groups? In particular they are finite sets... quite a few are too big to fit into your range, but the smallest (Mathieu groups) would do the trick.

http://en.wikipedia.org/wiki/Sporadic_groups

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As is well known, every finite (natural) number can be associated with a finite set of that cardinality. So in particular, the cardinality of a famous or special finite set must be a famous or special number. Here's a list of all the special numbers less than or equal to 9999 and contains quite a few items between $2^{10}$ and $2^{20}$.

Oh, what, you actually want the sets, and not just their cardinality, because there is more than one way of realising a set of a given cardinality? (Me grumbles something about bijective maps and isomorphisms of sets.) Fine:

  • 1132 is the number of 3-valent trees with 15 vertices
  • 1144 is the number of non-invertible knots with 12 crossings.
  • 1165 is the number of conjugacy classes in the automorphism group of the 12 dimensional hypercube.
  • 1205 is the number of fullerenes with 58 carbon atoms
  • 1294 is the number of 4 dimensional polytopes with 8 vertices.
  • 1378 is the number of symmetric idempotent 6×6 matrices over GF(2).
  • 1411 is the number of quasi-groups of order 5.
  • ...
  • 3240 is the number of 3×3×3 Rubik's cube positions that require exactly 3 moves to solve.
  • 3286 is the number of stable patterns with 16 cells in Conway's game of Life.
  • ...
  • 4535 is the number of unlabeled topologies with 7 elements.
  • ...

As beauty is in the eye of the beholder, I'm sure there are mathematicians out there who think each of the above numbers ought to be better known.

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