# Link between 24 dimension kissing number and Monster group

• A
Labyrinth
TL;DR Summary
Kissing number for 24 dimensions is 196560, monster group 196883.
I've heard that there is some link between these two values (they're so close!) but I can't seem to find it anywhere. Can someone point me in the right direction? (there's also the J-invariant 196884, well you get the idea)

Mentor
2022 Award
Which group are you talking about? The monster group has a several magnitude larger order.

Gold Member
Not that I have any clue, however, the prime factors

##196560 = 2^4 \times 3^3 \times 5 \times 7 \times 13##

##196883 = 47 \times 59 \times 71##

which have no common factors. Rules out homomorphism.

Edit: Okay, 196883 and 196560 aren't the group orders, so this isn't helpful.

Last edited:
Gold Member
I never understood the big deal. Is the monster group just the largest simple group?

Mentor
2022 Award
I never understood the big deal. Is the monster group just the largest simple group?
No. The largest of the 26 sporadic groups, but there are bigger regular ones, ##\mathbb{Z}_p## for example with an arbitrary large prime.

WWGD
The Monster group is much larger than you have said, and has ~8×10^53 elements.

Labyrinth
Which group are you talking about? The monster group has a several magnitude larger order.

The Monster group is the symmetry group of a 196,883 dimensional object. The value 196,884 appears as a coefficient in the Fourier expansion of the J-invariant.

These numbers are quite close to the kissing number in 24 dimensions: 196,560. I've heard that there is a connection but I can't quite find where this was all worked out, assuming it was.

Mentor
2022 Award
The Monster group is the symmetry group of a 196,883 dimensional object. The value 196,884 appears as a coefficient in the Fourier expansion of the J-invariant.

These numbers are quite close to the kissing number in 24 dimensions: 196,560. I've heard that there is a connection but I can't quite find where this was all worked out, assuming it was.
The connection is the Leech lattice. It determines the locations of the spheres for the 24-dimensional packing. Conway investigated the Leech lattice as he considered the groups.

https://de.wikipedia.org/wiki/John_Horton_Conway#Wirken said:
Conway discovered three new sporadic finite simple groups in the late 1960s, the Conway groups named after him, when he was studying the Leech lattice. He also simplified the construction of the last and largest sporadic group found, the "monster" (but preferred by the discoverer to be called "friendly giant"). In a famous work with his doctoral student Simon Norton from the late 1970s he pointed to the connections between the (dimensions of the irreducible) representations of the monster and the expansion coefficients of the elliptical module function, called "monstrous moonshine" after the title of their essay (she followed an observation by John McKay). Many of the suspected connections were later proven by Conway's PhD student Richard Borcherds, who received the Fields Medal for it. With his research group in Cambridge, Conway published the Atlas of Finite Groups.

WWGD