Some translation planes of order \(p^ 2\) and of extra-special type (Q1059280)

The main result is the following theorem. (a) There is a unique plane \(\pi\) which satisfies Hypothesis 1 (see below) and has kernel GF(p) for each of the primes \(p=3,5,7.\) (b) There are exactly four planes \(\pi\) which satisfy Hypothesis 1 and have kernel GF(11). Hypothesis 1. \(\pi\) is a translation plane of dimension 2 over its kernel GF(q). The translation complement of \(\pi\) contains a group G with the following property: G has a normal subgroup \(Q\cong Q_ 8*D_ 8\) and \(G/Q\cong A_ 5\).