The \((2,3)\)-generation of the special unitary groups of dimension 6 (Q2816946)

A \((2,3)\)-generated group is a group that can be generated by two elements of respective orders 2 and 3. Apart from the infinite families \(\mathrm{{PSp}}_4(2^a)\) and \(\mathrm{{PSp}}_4(3^a)\), the other finite classical simple groups are known to be \((2,3)\)-generated, up to a finite number of exceptions. A natural question is to detect all these exceptions. For \(n \leq 5\), the complete list of exceptions is known. In this paper, the authors closes the only case left open for \(n \leq 7,\) namely the 6-dimensional unitary groups. They give explicit \((2,3)\)-generators of the unitary groups \(\mathrm{{SU}}_6(q^2)\) for all \(q.\) These generators fit into a uniform sequence of likely \((2,3)\)-generators for all \(n \geq 6\).