Dihedral Group of Order 6 - Semidirect Products

Semidirect Products

is if both φ(0) and φ(1) are the identity. The semidirect product is isomorphic to the dihedral group of order 6 if φ(0) is the identity and φ(1) is the non-trivial automorphism of C₃, which inverses the elements.

Thus we get:

(n₁, 0) * (n₂, h₂) = (n₁ + n₂, h₂)

(n₁, 1) * (n₂, h₂) = (n₁ - n₂, 1 + h₂)

for all n₁, n₂ in C₃ and h₂ in C₂.

In a Cayley table:

00 10 20 01 11 21 00 00 10 20 01 11 21 10 10 20 00 11 21 01 20 20 00 10 21 01 11 01 01 21 11 00 20 10 11 11 01 21 10 00 20 21 21 11 01 20 10 00

Note that for the second digit we essentially have a 2x2 table, with 3x3 equal values for each of these 4 cells. For the first digit the left half of the table is the same as the right half, but the top half is different from the bottom half.

For the direct product the table is the same except that the first digits of the bottom half of the table are the same as in the top half.