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Out of interest, I've been reading some about sporadic groups and stumbled upon the Mathieu Groups, the smallest sporadic groups. My question is hopefully simple.

Since $M_{24}$ is often defined as the automorphism group of the Steiner system $S(5,8,24)$, is it equivalent to say that $M_{24}$ is the automorphism group on the Golay Code? Or is it the automorphism group of only the octads in the Golay Code?

Here, octads mean elements of the Golay Code with weight $8$.

Thank you!

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    As you say yourself, it depends on which definition you use. If you define it to be the automorphism group of $S(5,8,24)$ then this is equivalent to saying that it consists of those elements of $S_{24}$ that permute the octads of the Golay code. You would need to prove that all such elements are automorphisms of the Golay code. – 2017-02-23
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    So in other words, we discard the other elements of the Golay Code, save the octads? Then $M_{24}$ is the permutation group on the remaining objects? – 2017-02-23
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    It does follow that $M_{24}$ is also the group of automorphisms of the extended binary Golay code. The other words of that code are mod 2 sums of (the characteristic functions of) those octads. – 2017-02-24

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