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Macroscopic magnetic properties and microscopic magnetic structure ofRb$_2$Mn$_3$(MoO$_4$)$_3$(OH)$_2$ (space group $Pnma$) are investigated bymagnetization, heat capacity and single-crystal neutron diffractionmeasurements. The compound's crystal structure contains bond-alternating[Mn$_3$O$_{11}$]$^{\infty}$ chains along the $b$-axis, formed by isoscelestriangles of Mn ions occupying two crystallographically nonequivalent sites(Mn1 site on the base and Mn2 site on the vertex). These chains are only weaklylinked to each other by nonmagnetic oxyanions. Both SQUID magnetometry andneutron diffraction experiments show two successive magnetic transitions as afunction of temperature. On cooling, it transitions from a paramagnetic phaseinto an incommensurate phase below 4.5~K with a magnetic wavevector near ${\bfk}_{1} = (0,~0.46,~0)$. An additional commensurate antiferromagneticallyordered component arises with ${\bf k}_{2} = (0,~0,~0)$, forming a complexmagnetic structure below 3.5~K with two different propagation vectors ofdifferent stars. On further cooling, the incommensurate wavevector undergoes alock-in transition below 2.3~K. The experimental results suggest that themagnetic superspace group is $Pnma.1'(0b0)s0ss$ for the single-${\bf k}$incommensurate phase and is $Pn'ma(0b0)00s$ for the 2-${\bf k}$ magnetic phase.We propose a simplified magnetic structure model taking into account the majorordered contributions, where the commensurate ${\bf k}_{2}$ defines theordering of the $c$-axis component of Mn1 magnetic moment, while theincommensurate ${\bf k}_{1}$ describes the ordering of the $ab$-planecomponents of both Mn1 and Mn2 moments into elliptical cycloids

Complex Magnetic Order in a Decorated Spin Chain System Rb$_2$Mn$_3$(MoO$_4$)$_3$(OH)$_2$