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In a paper that I am reading, the author said that there exists a simple group of order 168 and it is known to have two distinct conjugacy classe of subgroups of order 24.

My question: Are they probably referring to $PSL(3,2)$? If so, what are the the subgroups of order 24 explicitly.

R Maharaj
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    Yes, it is $PSL(3,2)\simeq PSL(2,7)$. It is a non-obvious but somewhat standard exercise to prove that any simple group of order $168$ is isomorphic to the subgroup of $A_8$ that you get from a study of the action of $PSL(2,7)$ on the projective line $\Bbb{P}^1(\Bbb{F}_7)$. – Jyrki Lahtonen Mar 17 '17 at 11:02
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    Agreed, but I used to find it easier to construct the two classes of $S_4$s, and then use these to provide the projective plane over $\mathbb{F}_2$. – ancient mathematician Mar 17 '17 at 11:15

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As there is just one simple group of order 168, they are, and it is more easily seen to be $\text{GL}(3,2)$.

You will get one class of subgroups isomorphic to $S_4$ by taking the block matrices $$ \left[ \begin{matrix} A & 0\\ v&1\\ \end{matrix} \right] $$ where the $2\times 2$ matrix $A$ is of course non-singular.

These stabilise a [whatever] in the projective plane, and as I recall it the other class of $S_4$'s stabilise a [dual whatever].

ancient mathematician
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