We investigate the population of dwarf galaxies with stellar masses similar to the Large Magellanic Cloud (LMC) and M33 in the EAGLE galaxy formation simulation. In the field, galaxies reside in haloes with stellar-to-halo mass ratios of $1.03^{+0.50}_{-0.31}times10^{-2}$ (68% confidence level); systems like the LMC, which have an SMC-mass satellite, reside in haloes about 1.3 times more massive, which suggests an LMC halo mass at infall, $M_{200}=3.4^{+1.8}_{-1.2}times10^{11}M_odot$ (68% confidence level). The colour distribution of dwarfs is bimodal, with the red galaxies ($g-r>0.6$) being mostly satellites. The fraction of red LMC-mass dwarfs is 15% for centrals, and for satellites this fraction increases rapidly with host mass: from 10% for satellites of Milky Way (MW)-mass haloes to nearly 90% for satellites of groups and clusters. The quenching timescale, defined as the time after infall when half of the satellites have acquired red colours, decreases with host mass from ${>}5$ Gyrs for MW-mass hosts to $2.5$ Gyrs for cluster mass hosts. The satellites of MW-mass haloes have higher star formation rates and bluer colours than field galaxies. This is due to enhanced star formation triggered by gas compression shortly after accretion. Both the LMC and M33 have enhanced recent star formation that could be a manifestation of this process. After infall into their MW-mass hosts, the $g-r$ colours of LMC-mass dwarfs become bluer for the first 2 Gyrs, after which they rapidly redden. LMC-mass dwarfs fell into their MW-mass hosts only relatively recently, with more than half having an infall time of less than 3.5 Gyrs.
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