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 n isomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in one-to-one 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.
For many of us, when we think of coloring, we connect to the innocence of childhood and a time when we weren't burdened by responsibility and expectation. Butterflies are a great source for learning the letter ”B”, as well as learning about their metamorphosis from larvae, to caterpillar, to chrysalis, to adult butterfly. You can find many clip arts with slightest modifications yet with different subject matters either, thus there is a continuing process of preparing, producing and using of these school clip arts by a lot of people.