Elusive groups from non-split extensions

A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts that no elusive group is $2$-closed. Existing constructions of elusive groups mostly arise from split extensions. In this paper, we initiate the construction of elusive groups via non-split extensions. As a demonstration, we construct elusive groups of new degrees, namely $p^{3k-4}(p+1)/2$ for each Mersenne prime $p\geq7$ and integer $k\geq2$. We also construct the first examples of elusive groups with odd degree, namely $3^{k+1}\cdot5^2$, and twice odd degree, namely $2\cdot3^{k + 1}\cdot5^2$ for each $k\geq1$. We conclude by proposing further problems to advance this new direction of research.