It is a 4-fold transitive permutation group on 23 objects.
The Schur multiplier and the outer automorphism group are both trivial.
Milgram (2000) calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish.
The inverse Galois problem is solved for all other sporadic simple groups.
One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M21.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 24.A7.