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S6, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of nisomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in 1-1 correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n=6.
6 similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonallattice in which each circle touches just six others.
The evolution of our modern glyph for 6 appears rather simple when compared with that for the other numerals. Our modern 6 can be traced back to the Brahmins of India, who wrote it in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Ghubar Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.
On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a 'b' is not practical.
The sixth album by Dream Theater, Six Degrees Of Inner Turbulence, was based around the number six: the album has six songs, and the sixth song — that is, the complete second disc — explores the stories of six individuals suffering from various mental illnesses.
The National Basketball Association has six divisions (the National Hockey League had six divisions until realigning into four divisions for the 2013-14 season)
The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit. They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from 1942 to 1967
In rugby union, the starting blindside flanker wears jersey number 6. (Some teams use "left" and "right" flankers instead of "openside" and "blindside", with 6 being worn by the starting left flanker.)
The jersey number 6 has been retired by several North American sports teams in honor of past playing greats or other key figures (or, in two cases, a team's fans):
The Toronto Maple Leafs, for Hall of FamerAce Bailey. The Leafs have a unique policy of not retiring numbers unless the player honoured either died or suffered a career-ending incident while a member of the team. Bailey suffered a fractured skull during a game in 1933; while he recovered and lived for nearly 60 years after the incident, he never played again. The Leafs would issue the number to Ron Ellis in 1968 at Bailey's personal request, and Ellis wore it until his own retirement in 1981.
On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry it is the key for J and K, and on the BlackBerry 8700 series and Curve 8900 with full keyboard, it is the key for F