Sharply 5-Transitive: M12

The order of M12 is 95,040, which is 132 * 720

Since M12 is the automorphism group of S(5,6,12),
and is sharply 5-transitive, and maps blocks to blocks,
would there be 720 operations to go from say, hexad(A)
to hexad(B), (and B to C, etc) such that

x1, x2, x3, x4, x5 -> y1, y2, y3, y4, y5; and every combination

x1, x2, x3, x4, x5 -> (y's scrambled 5!) which makes 120,
and then times a factor of 6, because there are 6 pentads in each hexad, such that C6,1 = C6,5 pentads, to get every 720 g's going between two hexads? I sense it is more complex than this. I also am assuming all hexads are treated equally, which is probably wrong, such that
there are 132 cycles between hexads, going A,B,C,...
A,C,E... (and B,D,F..), A,D,G etc

Or does sharply transitive mean there is only one (not 120) going between g1 through g5 -> h1 through h5?

Rotman seems to be beyond my scope, even though I understand some of the theorems and proofs for this

PGH
.

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