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.