Exceptional groups of order 243

Ibrahim Alotaibi and David Easdown

Abstract

We describe all exceptional groups of order \(243=3^5\), with explanations and proofs, adjusting a table that appears in a 2017 paper by Britnell, Saunders and Skyner. There are ten exceptional groups of order \(243\), each of minimal degree \(18\), with four distinguished quotients, each of order \(81\) and minimal degree \(27\). Using a sieve technique, we identify all preimages of each distinguished quotient. The minimal degrees of the preimages become either (a) \(18\), when the preimage is exceptional, (b) \(27\), when the preimage is almost exceptional, (c) \(36\), or (d) \(54\). Cases (a), (c) and (d) occur with an elementary abelian centre of order \(9\), but with contrasting intersection properties using subgroups of order \(27\), leading to minimal representations afforded by two subgroups. Case (b) occurs with a cyclic centre of order \(3\) and a transitive minimal representation. We prove that there are exactly two nonisomorphic exceptional groups of order \(243\) having more than one (in fact two) nonisomorphic distinguished quotients.

Keywords: permutation groups, minimal degrees.

AMS Subject Classification: Primary 20B35.