The phrase “octad group” discussed here in a post
of March 7 is now a domain name, “octad.group,”
that leads to that post. Remarks by Conway and
Sloane now quoted there indicate how the group
that I defined in 1979 is embedded in the large
Mathieu group M24.
Related literary notes — Watson + Embedding.
From "Mathieu Moonshine and Symmetry Surfing" —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
"This presentation of the symmetry groups Gi is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group
G = (Z2)4 ⋊ A8 .
It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O9 = {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."
This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring," by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A-193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the "octad group" in 1993 —
Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.
This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M24.
* See a Log24 post of June 1, 2013.
The above title might describe the long damned nightmare
that is the history of the human species, or — a usage I prefer —
a concept from pure mathematics. For an example of the latter,
see posts tagged Octad Group and the URL http://octad.group.
The April 20 summary I wrote for ScienceOpen.com suggests
a different presentation of an Encyclopedia of Mathematics
article from 2013 —
(Click to enlarge.)
Keywords: PG(3,2), Fano space, projective space, finite geometry, square model,
Cullinane diamond theorem, octad group, MOG.
Cullinane, Steven H. (2021).
“The Square Model of Fano’s 1892 Finite 3-Space.”
Zenodo. https://doi.org/10.5281/zenodo.4718182 .
An earlier version of the square model of PG(3,2) —
Update on April 20, 2021 —
The following was added today to the above summary:
“It describes a group of 322,560 permutations, later known as
‘the octad group,’ that now plays a role in speculative high-energy physics.
See Moonshine, Superconformal Symmetry, and Quantum Error Correction .”
The "bricks" in posts tagged Octad Group suggest some remarks
from last year's HBO "Watchmen" series —
Related material — The two bricks constituting a 4×4 array, and . . .
"(this is the famous Kummer abstract configuration )"
— Igor Dolgachev, ArXiv, 16 October 2019.
As is this —
.
The phrase "octad group" does not, as one might reasonably
suppose, refer to symmetries of an octad (a "brick"), but
instead to symmetries of the above 4×4 array.
A related Broomsday event for the Church of Synchronology —
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