Context. Early-type stars have convective cores due to a steep temperature gradient produced by the CNO cycle. These cores can host dynamos and the generated magnetic fields may be relevant in explaining the magnetism observed in Ap/Bp stars. Aims. Our main objective is to characterise the convective core dynamos and differential rotation. We aim to carry out the first quantitative analysis of the relation between magnetic activity cycle and rotation period. Methods. We used numerical 3D star-in-a-box simulations of a 2.2 M-circle dot A-type star with a convective core of roughly 20% of the stellar radius surrounded by a radiative envelope. We explored rotation rates from 8 to 20 days and used two models of the whole star, along with an additional zoom set where 50% of the radius was retained. Results. The simulations produce hemispheric core dynamos with cycles and typical magnetic field strengths around 60 kG. However, only a very small fraction of the magnetic energy is able to reach the surface. The cores have solar-like differential rotation and a substantial part of the radiative envelope has a quasi-rigid rotation. In the most rapidly rotating cases, the magnetic energy in the core is roughly 40% of the kinetic energy. Finally, we find that the magnetic cycle period, P-cyc, increases with decreasing the rotation period, P-rot, which has also been observed in many simulations of solar-type stars. Conclusions. Our simulations indicate that a strong hemispherical core dynamo arises routinely, but that it is not enough the explain the surface magnetism of Ap/Bp stars. Nevertheless, since the core dynamo produces dynamically relevant magnetic fields, it should not be neglected even when other mechanisms are being explored.