# Finite groups of order 24 in a special unitary Lie group

Here is a list of finite groups of order 24: https://groupprops.subwiki.org/wiki/Groups_of_order_24

There are 15 of them.

What constraints and methods can be used to check, among 15 of them, are there some of them can be a subgroup of Lie group SU(3)?

Then, what are these groups of order 24 that are the subgroup of Lie group SU(3)?

In one line : look at the character table of your $15$ groups of order $24$. To be a little more accurate:

• a finite group $G$ is isomorphic to a subgroup of $SU(3)$ if and only if it has faithful representation in $\mathbb{C}^3$,

• any representation of $G$ on $\mathbb{C}^3$ will be decomposed as a sum of $3$ $1$-dimensional representations, a sum of $1$-dimensional representation and a $2$-dimensional irreducible representation or will be an irreducible representation,

• if you are given a representation $\rho$ of a finite group $G$ on $\mathbb{C}^d$ then $g\in \ker \rho$ if and only if $tr\rho(g)=d$.

As a result, you can easily construct an algorithm that, given the character table of a group determines if your group is isomorphic to a subgroup of $SU(3)$.

Remarks:

• this works in any dimension, for any finite group,

• by being slightly more careful I think you can get a conjugacy class classification instead of a isomorphism class classification,

• Among these groups you will probably find $S_4$, $D_{12}$ and $\mathbb{Z}/24$.

• @ Clément Guérin, thanks for the nice answer +1! If you are sure that the subgroups are the three written above and no more among the 15 groups, then I can accept it as an answer right away. Or, I will wait a few days... – wonderich Apr 8 '18 at 23:32
• @wonderich, you are welcome. You should wait a few days, the list is not meant to be exhaustive. – Clément Guérin Apr 9 '18 at 0:20
• @ Clément Guérin, How about this, do you know this as well? math.stackexchange.com/questions/2728705/… – wonderich Apr 9 '18 at 2:27