Multiplication Table for the Permutation Group S4
A color-coded example of non-trivial abelian, non-abelian, and normal subgroups, quotient groups and cosets.

This image shows the multiplication table for the permutation group S4, and is helpful for visualizing various aspects of groups. This group consists of all the permutations possible for a sequence of four numbers, and has 24 (= 4!) elements. For example, the element r1 rotates the sequence one place to the left. That is, if the original sequence looks like (1234) then r1 maps (1234) -> (2341). The mirror element m0 flips the order, i.e. (1234) -> (4321). The combination of m0*r1 first rotates, then mirrors, so the map is (1234) -> (1432), which is defined in the Key as element m1, so m0*r1 = m1.

The elements in the Key are defined as follows. e is the identity element, and the r's are rotations that shift values from right to left. The element m0 is a mirror, with m1 = m0*r1 = (1432) a rotation followed by a mirror, and similarly for m2 and m3. The elements a0, b0, c0, and d0 are simple transpositions; for example, a0 flips the first and second element so a0 = (2134). a1 rotates with r1 and then applies a0, and similarly for a2 and a3. The other transpositions b, c, and d act similarly.

There are many things to notice here.