S5 S5 is the first non-solvable symmetric group. Along with the special linear group and the icosahedral group , S5 is one of the three non-solvable groups of order 120, up to isomorphism. S5 is the Galois group of the general quintic equation, and the fact that S5 is not a solvable group translates into the non-existence of a general formula to solve quintic polynomials by radicals. There is an exotic inclusion map as a transitive subgroup; the obvious inclusion map fixes a point and thus is not transitive. This yields the outer automorphism of S6, discussed below, and corresponds to the resolvent sextic of a quintic. S6 Unlike all other symmetric groups, S6, has an outer automorphism. Using the language of Galois theory, this can also be understood in terms of Lagrange resolvents. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map as a transitive subgroup (the obvious inclusion map fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S6—see Automorphisms of the symmetric and alternating groups for details. Note that while A6 and A7 have an exceptional Schur multiplier (a triple cover) and that these extend to triple covers of S6 and S7, these do not correspond to exceptional Schur multipliers of the symmetric group. Maps between symmetric groups Other than the trivial map and the sign map , the most notable homomorphisms between symmetric groups, in order of relative dimension, are: corresponding to the exceptional normal subgroup ; (or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of S6. as a transitive subgroup, yielding the outer automorphism of S6 as discussed above. There are also a host of other homomorphisms where . Relation with alternating group For , the alternating group An is simple, and the induced quotient is the sign map: which is split by taking a transposition of two elements. Thus Sn is the semidirect product , and has no other proper normal subgroups, as they would intersect An in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in An (and thus themselves be An or Sn). Sn acts on its subgroup An by conjugation, and for , Sn is the full automorphism group of