  A minimal Kitaev-Gamma model has been recently investigated to understand
various Kitaev systems. In the one-dimensional Kitaev-Gamma chain, an emergent
SU(2)$_1$ phase and a rank-1 spin ordered phase with $O_h\rightarrow D_4$
symmetry breaking were identified using non-Abelian bosonization and numerical
techniques. However, puzzles near the antiferromagnetic Kitaev region with
finite Gamma interaction remained unresolved. Here we focus on this parameter
region and find that there are two new phases, namely, a rank-1 ordered phase
with an $O_h\rightarrow D_3$ symmetry breaking, and a peculiar Kitaev phase.
Remarkably, the $O_h\rightarrow D_3$ symmetry breaking corresponds to the
classical magnetic order, but appears in a region very close to the
antiferromagnetic Kitaev point where the quantum fluctuations are presumably
very strong. In addition, a two-step symmetry breaking $O_h\rightarrow
D_{3d}\rightarrow D_3$ is numerically observed as the length scale is
increased: At short and intermediate length scales, the system behaves as
having a rank-2 spin nematic order with $O_h\rightarrow D_{3d}$ symmetry
breaking; and at long distances, time reversal symmetry is further broken
leading to the $O_h\rightarrow D_3$ symmetry breaking. Finally, there is no
numerical signature of spin orderings nor Luttinger liquid behaviors in the
Kitaev phase whose nature is worth further studies.
