Preliminary Analysis of S(5,6,12) hexachords

Well,

This is what I have so far (which took a while, I might say)
Out of the 132 Steiner hexads in S(5,6,12) I analyzed the
distribution of the 80 hexachord types (924 hexachords reduced for
transposition)

16 are not used at all (this is 1/5 of the 80 hexachords)
20 are used once (one transposition)
28 are used twice (two transpositions)
12 are used three times (three transpositions)
2 are used four times (four transpositions)
2 are used five times (five transpositions)

Multiply this out and - I know - you get 130 hexachords (the site I
got this from is missing two). If anyone knows where to find *valid*
listings of hexads in S(5,6,12), S(3,6,22) etc. Please let me know -
I haven't been successful (hunted Lecture notes in Math, etc.)

A quick summary -

S(5,6,12) is the Steiner system (a listing of 132 hexads) where *all*
pentachords in 12-et (792 of them) each fit into exactly one hexad.
Six different pentachords fit into one hexad, so you get 132*6=792
pentachords. It's also interesting that every *note* appears exactly
66 times - 12*66 also equals 132*6, the number of notes in the 132
hexad blocks. The Mathieu group based on this system has order 95040
which is P(12,5). More on S(3,6,22) and 22-et later!

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:

> If anyone knows where to find *valid*
> listings of hexads in S(5,6,12), S(3,6,22) etc. Please let me know -
> I haven't been successful (hunted Lecture notes in Math, etc.)

John McKay sent me an octad list for M24, from which these should be
extractable. Unfortunately, my modem broke and I can't get my
replacement to install, so I'm waiting for by brother to get back to
the US and meanwhile using computers in the library.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
>
> > If anyone knows where to find *valid*
> > listings of hexads in S(5,6,12), S(3,6,22) etc. Please let me
know -
> > I haven't been successful (hunted Lecture notes in Math, etc.)
>
> John McKay sent me an octad list for M24, from which these should be
> extractable. Unfortunately, my modem broke and I can't get my
> replacement to install, so I'm waiting for by brother to get back to
> the US and meanwhile using computers in the library.

Thanks! I have been exploring the properties of M22, M23 and M24...