An alternating group is a group of even permutations on a set of length n, denoted A_n or Alt(n). Alternating groups are therefore permutation groups. The nth alternating group is represented in the Wolfram Language as AlternatingGroup[n]. An alternating group is a normal subgroup of the permutation group, and has group order n!/2, the first few values of which for n = 2, 3, ... are 1, 3, 12, 60, 360, 2520, ... (OEIS A001710). The alternating group A_n is (n - 2)-transitive. Amazingly, the pure rotational subgroup I of the icosahedral group I_h is isomorphic to A_5. The full icosahedral group I_h is isomorphic to the group direct product A_5×C_2, where C_2 is the cyclic group on two elements.
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