2-bit Walsh permutation

There are Sloane'sA002884(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).

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Cayley table

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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