V446 Cephei: a $\beta$ Cep pulsator in a multiple system

$\beta$ Cep stars in eclipsing binary (EB) systems give us an opportunity to put observational constraints on their structure and stellar parameters. We present a comprehensive analysis of the $\beta$ Cep star in the EB V446 Cep, using \textit{TESS} photometry and HERMES spectra. We calculate the stellar and orbital parameters using light curve modelling and spectral disentangling. The EB has an orbital period of $3.808567 \pm 0.000012$ d and a mass ratio of $0.1550 \pm 0.0012$. We find the $\beta$ Cep star to have a mass of $10.68 \pm 0.06$ $\mathrm{M_{\odot}}$, a radius of $5.864 \pm 0.033$ $\mathrm{R}_{\odot}$, and a $T_{\rm eff}$ of $24220 \pm 180$ K. The secondary has a mass of $1.657 \pm 0.017$ $\mathrm{M_{\odot}}$, a radius of $1.530 \pm 0.014$ $\mathrm{R}_{\odot}$, and a $T_{\rm eff}$ of $9080 \pm 390$ K. We also extract the abundances of C, N, O, Mg, and Si for the $\beta$ Cep star, which are found to be consistent with galactic OB binaries. We identified 21 distinct pulsation frequencies, with the dominant mode at 10.24324 d$^{-1}$, which corresponds to a near-harmonic of the system's orbital frequency. The two stars in the EB have asynchronous rotation, with both stars rotating faster than the orbital frequency. We detect a companion to the EB using eclipse timing variations and period changes of the dominant pulsation frequency. We calculate the minimum mass of this tertiary companion to be $4.11 \pm 0.32$ $\mathrm{M_{\odot}}$ which is on an orbit of 2303$\pm$69 d around the EB. Using spectral energy distributions and MIST isochrones, we conclude that V446 Cep is either a co-evolving hierarchical 2+2 quadruple or a triple system where the third body is a compact object.