The collineation groups of the translation planes of order 25 (Q809349)

The author gives an efficient method to determine collineation groups of translation planes of order \(q^ 2\), defined by spread sets. More precisely, a computer program is given which establishes the action \(\bar G\) of the collineation group G of a plane \(\pi\) at the line at infinity of \(\pi\) (under assumption that either points (0) and (\(\infty)\) are in the same G-orbit, or (\(\infty)\) is fixed). Exact data in the case \(q=5\) are presented, concerning orders and orbit lengths for groups of all the translation planes of order \(q^ 2\). These planes were (also efficiently) found by Oakden and the author; there are seven of them which have not previously appeared in the literature, and the structure of their collineation group is explained in more details.