Affine group in Z12

Would someone explain the Affine group in Z12?

I think I am close, and BTW

C12 = D4+ X S3+ = C4 X C3 = M1 transform
D12 = D4- X S3- = M11 transform
D4- X S3+ = D4- X C3 = M7 transform
D4+ X S3- = C4 X S3- = M5 transform

So, is D4 X S3 related to the Affine Group?

If someone answers this, it would be appreciated. And I might not be
so stuck on 12tET.

Of course, the Z-relation in 12-tET is related to my S2C operator
(S2-complementability)

All these together get hexachords down to 26.

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:
>
> Would someone explain the Affine group in Z12?

The group of units is +-1, +-5 mod 12. This is, under multiplication
mod 12, the Klein 4-group. The full affine group is therefor a
semidirect product of this with C12, of order 48.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@>
> wrote:
> >
> > Would someone explain the Affine group in Z12?
>
> The group of units is +-1, +-5 mod 12. This is, under multiplication
> mod 12, the Klein 4-group. The full affine group is therefor a
> semidirect product of this with C12, of order 48.
>
Thanks. That falls in line with what I seem to understand but it is
nice to have it validated. I assume also that the Klein Vieresgruppe
is isomorphic with C2 X C2, which follows from the Multiplicatve
Modulo stuff.