Computing labels of subgroups (awaiting review)

In order to label Gassmann classes we first fix an ordering of the conjugacy classes of $G$. Having fixed such an ordering, it is easy to compute a unique identifier for each Gassmann class: we count the intersection of $H$ with each $G$-conjugacy class. We enumerate $H$-conjugacy classes of elements of $H$ in any order, then determine which $G$-conjugacy class they belong to, adding the size of the $H$-conjugacy class to a running tally of the intersection. An analogous process works for Gassman vectors up to automorphism, where we collect together $G$-conjugacy classes that are related by an automorphism of $G$.

If there are multiple subgroups up to automorphism (resp., conjugation) within a given Gassmann class, we use the subgroup lattice to further order those remaining subgroups. We first order subgroups $H$ in the same Gassmann class using the lex ordering on the sorted list of labels of all (proper) supergroups of $H$. (One can and often does have Gassmann equivalent subgroups for which the lists of supergroups differ.) Note that here “supergroup” refers to inclusions in the poset of subgroups up to automorphism (resp., conjugation), meaning we consider $K$ to be a supergroup of $H$ if it contains any subgroup equivalent to $H$ up to automorphism (resp., conjugation).

In these cases where two or more Gassmann equivalent subgroups have the same set of supergroups, we resort to computing permutation representation signatures. We can, in fact, compute with any supergroup $K$ of $H$ instead of $G$. This is much more efficient when the index $[K:H]$ is much smaller than $[G:H]$, but does require some care to account for the fact that there may be two or more $G$-conjugates of $H$ contained in $K$ that are not $K$-conjugate.