How to prove that a given group is isomorphic to Sym(4)?

Question

Given a specific group with 24 elements, I want to prove that it is isomorphic to Sym(4).

To begin with, I calculate the orders of my group's elements and they come out as in the order statistics for Sym(4): 1 of order 1, 9 of order 2, 8 of order 3 and 6 of order 4. Now as I understand, matching the elements orders is not enough to show isomorphism for non-abelian groups. But I was wondering if in this particular case it happens to be enough, since there are only 15 groups of order 24. See the groupprops database. Unfortunately, this database does not list the order statistics for all the other groups of order 24 so I can rule them out. Is there any other place where I can find their order statistics?

If the above method does not work, then I'm thinking to show isomorphism as follows: my group is a set of 24 elements and it also acts on various sets of elements. I can take a specific set with 4 elements and show by direct calculations that my group acts on it by permuting these 4 elements. I do indeed get all the 24 permutations. Would this be enough to prove isomorphism to Sym(4)? If so, what well-known theorem can I quote in support? (I only know the very basics of group theory.)

Answer 1

In fact in this case the order statistics do give you enough information to prove that the group is isomorphic to $S_4$ but I agree with Jack Yoon that this may not be the best approach.

A group of order $24$ has $1$ or $4$ Sylow $3$-subgroups, and the fact that there are $8$ elements of order $3$ shows that there must be $4$.

The image $P$ of the conjugation action of $G$ on the set of its Sylow $3$-subgroups is transitive, with a Sylow $3$-subgroup fixing a unique point, so $P$ contains $A_4$.

If $P = A_4$ then $G$ would have a normal and hence unique Sylow $2$-subgroup, in which case there would be just $7$ elements of order $2$ or $4$, which is not the case. Hence $P=S_4$ and $P \cong G$.

Answer 2

After reading the first part, I was just about to suggest the method of your last paragraph. This is the preferred method to show the desired result, I suppose. For example, the symmetry group of a cube has $24$ elements because we can pick a face and for this face four different orientations. The fact that the group is isomorphic to $S_4$ becomes immediately apparent when one notices that the group acts faithfully(!) on the four-element set of spacial diagonals.