The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8-set into two 4-sets and the 35 lines in the
finite projective space PG(3,2).
Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2-subsets of a 6-set and the 15 points of PG(3,2).
An analogy between the two above analogies
is supplied by the exceptional outer automorphism of S6.
See…
The 2-subsets of a 6-set are the points of a PG(3,2),
Picturing outer automorphisms of S6, and
A linear complex related to M24.
(Background: Inscapes, Inscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3-space.)
* For some context, see Analogies and
"Smallest Perfect Universe" in this journal.