**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 *G _{i}* 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* = (Z_{2}**)**^{4}^{ }**⋊**** A**_{8}_{ }.

It can be described as a maximal subgroup of M_{24}

obtained by the setwise stabilizer of a particular

‘reference octad’ in the Golay code, which we take

to be* O*_{9 }= {3,5,6,9,15,19,23,24} ∈ 𝒢_{24}. The octad

subgroup is of order 322560, and its index in M_{24}

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 —