Abstract
We construct conformally scalar NUTty dyon black holes in non-linear electrodynamics (NLE) which possess conformal and duality-rotation symmetries and are characterized by a free dimensionless parameter. The thermodynamic first law of the black hole is formulated. Then we explore the strong gravitational effects of the black hole, mainly focusing on the innermost stable circular orbits (ISCOs) of massive particles and shadows formed by the photons. One of our interesting results is that if the black hole is endowed with a positive real conformal scalar charge rendering it Reissner-Nordstr\"om-like, the radii of the ISCOs and the shadow both increase with the increasing NLE parameter signifying the increasing nonlinearity of the electromagnetic field.
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