Finite primitive permutation groups admitting a quasi-semiregular element have O'Nan-Scott type HA, AS, PA, SD or CD, and the alternating and sporadic almost simple cases are classified explicitly.
Breuer, The GAP Character Table Library, Version 1.3.9, GAP Package, https://www
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Finite permutation groups with quasi-semiregular elements
Finite primitive permutation groups admitting a quasi-semiregular element have O'Nan-Scott type HA, AS, PA, SD or CD, and the alternating and sporadic almost simple cases are classified explicitly.