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.