2-bit Walsh permutation

There are  A002884(2) = 2 * 3 = 6 invertible binary 2×2 matrices.

They form the general linear group GL(3,2). It is isomorphic to the symmetric group S₃ and the dihedral group D₃ (see here).

overview		transformation arrows		transformed object
+	−	+	−	+	−

transformed objects in two directions
vector	det.	arrows	as inverse	binary
1 2	+
2 1	−
1 3	+
3 1	−
3 2	+
2 3	−

Cayley table

2×2 transformation matrices	2×2 transformation and 4×4 permutation matrices	all 3×3 permutation matrices (same as 4×4 matrices without trivial fixed point)
one of the S3 subgroups of S4