The Exceptional Outer Automorphism of S6
Among symmetric groups, only S6 has a (non-trivial) outer automorphism, which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S6) = C2.
This was discovered by Otto Hölder in 1895.
This also yields another outer automorphism of A6, and this is the only exceptional outer automorphism of a finite simple group: for the infinite families of simple groups, there are formulas for the number of outer automorphisms, and the simple group of order 360, thought of as A6, would be expected to have 2 outer automorphisms, not 4. However, when A6 is viewed as PSL(2, 9) the outer automorphism group has the expected order. (For sporadic groups (not falling in an infinite family), the notion of exceptional outer automorphism is ill-defined, as there is no general formula.)
Read more about this topic: Automorphisms Of The Symmetric And Alternating Groups
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