Magnetization measurements on single-crystal cubic SrCuTe$_2$O$_6$ with an applied magnetic field of along three inequivalent high symmetry directions $[100]$, $[110]$, and $[111]$ reveal weak magnetic anisotropy. The fits of the magnetic susceptibility to the result from a quantum Monte Carlo simulation on the Heisenberg spin-chain model, where the chain is formed via the dominant third-nearest-neighbor exchange interaction $J_3$, yield the intra-chain interaction $(J_3/k_B)$ between 50.12(7) K for the applied field along $[110]$ and 52.5(2) K along $[100]$ with about the same $g$-factor of 2.2. Single-crystal neutron diffraction unveils the transition to the magnetic ordered state as evidenced by the onset of the magnetic Bragg intensity at $T_textrm{N1}=5.25(9)$ K with no anomaly of the second transition at $T_textrm{N2}$ reported previously. Based on irreducible representation theory and magnetic space group analysis of powder and single-crystal neutron diffraction data, the magnetic structure in the Shubnikov space group $P4_132$, where the Cu$^{2+}$~$S=1/2$ spins antiferromagnetically align in the direction perpendicular to the spin chain, is proposed. The measured ordered moment of $0.52(6)~mu_B$, which represents 48% reduction from the expected value of $1~mu_B$, suggests the remaining influence of frustration resulting from the $J_1$ and $J_2$ bonds.
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