= Equality-pattern orbits of projective triples
{title2=$1+3+1\text{ orbit types on }(\mathbb P^1(\mathbb F_p))^3$}
The <general linear group over a finite field> acting diagonally on projective triples has five <orbits of a group action>: all equal, three different positions for the repeated coordinate when exactly two are equal, and all distinct. The corresponding representative <stabilizer subgroups> are upper triangular, diagonal, and <scalar matrices>. Their orders are $p(p-1)^2$, $(p-1)^2$, and $p-1$, respectively; the orbit sizes are $p+1$, $p(p+1)$, and $p(p+1)(p-1)$. Transitivity follows from the action being <sharply three-transitive on a projective line>.
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