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The normaliser problem takes as input subgroups $G$ and $H$ of the symmetricgroup $S_n$, and asks one to compute $N_G(H)$. The fastest known algorithm forthis problem is simply exponential, whilst more efficient algorithms are knownfor restricted classes of groups. In this paper, we will focus on groups withmany orbits. We give a new algorithm for the normaliser problem for thesegroups that performs many orders of magnitude faster than previousimplementations in GAP. We also prove that the normaliser problem for thespecial case $G=S_n$ is at least as hard as computing the group of monomialautomorphisms of a linear code over any field of fixed prime order.

Computing normalisers of intransitive groups