# Groups of order 2024

This time last year I wrote about groups of order 2023 and now I’d like to do the same for 2024.

There are three Abelian groups of order 2024, and they’re not hard to find.

We can factor

2024 = 8 × 11 × 23

and so the Abelian groups of order 2024 are of the form

G ⊕ ℤ11 ⊕ ℤ23

where G is a group of order 8, and there are three possibilities for G:

• 8,
• 4 ⊕ ℤ2, and
• 2 ⊕ ℤ2 ⊕ ℤ2.

How many non-Abelian groups of order 2024 are there? Conway’s estimate would be a total of 52 groups, Abelian and non-Abelian, but that turns out to be a bit high.

There is a formula for the number of groups of order n, but it only applies to square-free numbers. The number 2024 is divisible by 4, so it’s not square-free.

There are 46 groups of order 2024, so 43 of these are non-Abelian. When n is divisible by a square, finding the number of groups of order n is hard, but the results have been tabulated for small n. I’ve seen a table going up to 2048 and no doubt there are tables that go further.

## One thought on “Groups of order 2024”

1. Paul Fabel

Are there any examples of a group G_2024 which is NOT a semidirect product of two of its subgroups? For example, G_23 is normal in G_2024 by Sylow(s), but there exist two groups G_253 of order 11 x 23, since 11 divides 22= 23-1.

Thus there exist several groups G_2024 as a semidirect product determined by a G_8–>AUT(G_253).

But which, if any, groups G_2024 are NOT a semidirect product determined by a homomorphism G_88—>AUT(G_23)?