  Umbral moonshine describes an unexpected relation between 23 finite groups
arising from lattice symmetries and special mock modular forms. It includes the
Mathieu moonshine as a special case and can itself be viewed as an example of
the more general moonshine phenomenon which connects finite groups and
distinguished modular objects. In this paper we introduce the notion of
generalised umbral moonshine, which includes the generalised Mathieu moonshine
[Gaberdiel M.R., Persson D., Ronellenfitsch H., Volpato R., Commun. Number
Theory Phys. 7 (2013), 145-223] as a special case, and provide supporting data
for it. A central role is played by the deformed Drinfel'd (or quantum) double
of each umbral finite group $G$, specified by a cohomology class in
$H^3(G,U(1))$. We conjecture that in each of the 23 cases there exists a rule
to assign an infinite-dimensional module for the deformed Drinfel'd double of
the umbral finite group underlying the mock modular forms of umbral moonshine
and generalised umbral moonshine. We also discuss the possible origin of the
generalised umbral moonshine.

Representation Theory;High Energy Physics - Theory;Number Theory