There's a slightly eccentric (but entertaining) book by Budden called "The Fascination of Groups" that has an extensive chapter about the application of groups to the British practice of "change ringing" (this actually is featured prominently in the Dorothy Sayers, Lord Peter Wimsey mystery, "The Nine Tailors"). Change Ringing is the practice of ringing all the permutations of some number of church bells (usually 5 or 6, but can be as many as 9) with the requirement that successive permutations differ by a two-cycle (i.e. just swapping the order of two particular bells). Why this practice arose, I don't really know, but it sets up interesting algorithmic and mathematical problems.
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