The largest of the
unclassifiable simple groups and therefore the final nail in the
longest proof in mathematics. It is of
order 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 (that's about 8 × 10
53). The prime factorisation of this order is 2
46·3
20·5
9·7
6·11
2·13
3·17·19·23·29·31·41·47·59·71.
Noether has pointed out to me that it is, if you're going to be strict and pedantic about it, sporadic rather than unclassifiable, since the classification includes the brute fact that there exist sporadic groups, not otherwise classifiable. It is also known as the monster group.
No, I don't really know what it means, beyond the fact that groups can be decomposed into the equivalent of prime numbers, and this is the strangest of them. But it caught my fancy. Well, I suppose it means that mathematics of the most fundamental kind can be very surprisingly irregular-looking. Astonishingly, the monster simple group has found a use in physics, being connected with 26-dimensional strings and complex tori.