Symmetric group S_6 - Knowlpedia

The symmetric group $S_6$ is the group of all bijections of $\{1,\ldots,6\}$ , with composition as multiplication.

Order and normal subgroup

There are 720 permutations: choose the images of the 6 letters successively. The even permutations form the proper nontrivial normal subgroup $A_6$ , the kernel of the sign map. Therefore $S_6$ is not simple.

References

J. S. Milne, Group Theory, v4.01, §4, symmetric and alternating groups; Theorem 4.33 and Remark 4.34, p. 69.