Alternating Groups that are Ambivalent

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Let $n$ be a natural number.

Then the $n$th alternating group $A_n$ is ambivalent if and only if $n \in \set {1, 2, 5, 6, 10, 14}$.

This sequence is A115200 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).