There are no simple groups of order $264$

Before this writing the smallest order for which the nonexistence of simple groups of that order is not explicitly demonstrated on this site is $$264$$; this self-answered question aims to fill that gap. (See Prove there are no simple groups of even order $<500$ except orders $2$, $60$, $168$, and $360$., the sole answer of which describes an outline for proving the titular statement but does not describe how to establish the claim for order $$264$$.) Other answers are, of course, welcome.

How does one show that there are no simple groups of order $$264$$?

• Before anyone down- or close-votes this for a seeming lack of attempts, check the answers! Jun 22, 2023 at 0:32
• Thanks, @Shaun . I made more explicit that the question is self-answered, to help ward off the issue you mention. Jun 22, 2023 at 0:52
• I rarely down or close vote, relax
– mick
Jun 22, 2023 at 1:30

This approach is essentially the one given in $$\S$$6.2 of Dummit & Foote's Abstract Algebra. A closely related approach is given in $$\S$$3 the cited article (as far as I know, the latter is the earliest proof of the claim for this order).
Suppose $$G$$ were simple of order $$264 = 2^3 \cdot 3 \cdot 11$$. Sylow's Theorems imply that $$n_{11} = 1 \pmod {11}$$ and $$n_{11} \mid 264$$; simplicity imposes $$n_{11} > 1$$ leaving $$n_{11} = 12$$ as the only possibility. So, we can identify $$G$$ with a subgroup of $$S_{12}$$, and since $$G$$ is simple, it contains no subgroup of index $$2$$, hence $$G \leq A_{12}$$. Now, let $$P$$ be a Sylow-$$11$$ subgroup; Frattini's Argument gives that $$|N_{A_{12}}(P)| = \frac{1}{2} |N_{S_{12}}(P)| = \frac{1}{2} (11)(11 - 1) = 55 .$$ But $$|N_G(P)| = \frac{|G|}{n_{11}} = \frac{264}{12} = 22$$, and $$22 \not\mid 55$$, which contradicts the fact that $$N_G(P) \leq N_{A_{12}}(P)$$.
Remark Cf. this answer, which uses a similar technique to resolve the case of order $$336$$.
F.N. Cole, "Simple Groups from Order $$201$$ to Order $$500$$," Amer. J. Math. 14(4) (October 1892), pp. 378$$-$$388.