= Diagonal primitive permutation group
{title2=$n=|T|^{r-1},\quad T^r\cap G_\alpha=\operatorname{diag}(T)$}
For a nonabelian simple group $T$, act on cosets of its full diagonal in $T^r$. A suitable factor-permuting extension is primitive and has point stabilizer intersecting the socle in the diagonal. The full normalizer has order $|T|^r|\operatorname{Out}(T)|r!$. For $T=A_5,r=2$, it acts on the sixty elements of $A_5$ by left/right multiplication, automorphisms and inversion, and has order $14400$. Odd conjugation in $S_5$ induces an odd permutation of these sixty points, while inversion is even; the full group is a primitive maximal subgroup of $S_{60}$.
Back to article page
