We determine, by a computer search, all the cyclic parallelisms of PG(3, 5). There are 45 of them, up to projective equivalence. In particular, there are two cyclic regular parallelisms of PG(3, 5). Previously, no example was known of a regular parallelism in PG(3, q), for q odd.

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The problem of determining two-dimensional translation planes admitting SL(2,5) in the translation complement has been intensively studied. Most examples arise from multiple-derivation of a Desarguesian plane. In this paper, we construct two translation planes of order 19 2 admitting SL(2,5), one of which is obtained by 12-nest replacement.