A Galois group that is neither $S_n$ nor $A_n$

A concrete example is the splitting field of \(x^4-2\) over \(\mathbb Q\). If \(\alpha=\sqrt[4]{2}\), its splitting field is \(K=\mathbb Q(\alpha,i)\), of degree \(8\). The automorphisms \[ r(\alpha)=i\alpha,\quad r(i)=i, \qquad s(\alpha)=\alpha,\quad s(i)=-i \] satisfy \[ r^4=s^2=1,\qquad srs=r^{-1}. \] Hence \[ \operatorname{Gal}(K/\mathbb Q)\cong D_4, \] the dihedral group of order \(8\). Since no symmetric or alternating group has order \(8\), this is neither \(S_n\) nor \(A_n\) for any \(n\).

Many other examples exist; for instance \(\operatorname{Gal}(\mathbb Q(\zeta_7)/\mathbb Q)\cong(\mathbb Z/7\mathbb Z)^\times\cong C_6\).