The overlap between the spectroscopic Galactic Archaeology with HERMES (GALAH) survey & $Gaia$ provides a high-dimensional chemodynamical space of unprecedented size. We present a first analysis of a subset of this overlap, of 7066 dwarf, turn-off, & sub-giant stars. [...] We investigate correlations between chemical compositions, ages, & kinematics for this sample. Stellar parameters & elemental abundances are derived from the GALAH spectra with the spectral synthesis code SME. [...] We report Li, C, O, Na, Mg, Al, Si, K, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn, Y, as well as Ba & we note that we employ non-LTE calculations for Li, O, Al, & Fe. We show that the use of astrometric & photometric data improves the accuracy of the derived spectroscopic parameters, especially $log g$. [...] we recover the result that stars of the high-$alpha$ sequence are typically older than stars in the low-$alpha$ sequence, the latter spanning $-0.7<$[Fe/H]$<+0.5$. While these two sequences become indistinguishable in [$alpha$/Fe] vs. [Fe/H] at the metal-rich regime, we find that age can be used to separate stars from the extended high-$alpha$ & the low-$alpha$ sequence even in this regime. [...] we find that the old stars ($>8$ Gyr have lower angular momenta $L_z$ than the Sun, which implies that they are on eccentric orbits & originate from the inner disk. Contrary to some previous smaller scale studies we find a continuous evolution in the high-$alpha$-sequence up to super-solar [Fe/H] rather than a gap, which has been interpreted as a separate high-$alpha$ metal-rich population. Stars in our sample that are younger than 10 Gyr, are mainly found on the low $alpha$-sequence & show a gradient in $L_z$ from low [Fe/H] ($L_z>L_{z,odot}$) towards higher [Fe/H] ($L_z<L_{z,odot}$), which implies that the stars at the ends of this sequence are likely not originating from the close solar vicinity.
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