  Like a three-qubit Greenberger-Horne-Zeilinger(GHZ) symmetry we explore a
corresponding symmetry in the four-qubit system, which we call GHZ$_4$
symmetry. While whole GHZ-symmetric states can be represented by two real
parameters, the whole set of the GHZ$_4$-symmetric states is represented by
three real parameters. In the parameter space all GHZ$_4$-symmetric states
reside inside a tetrahedron. We also explore a question where the given SLOCC
class of the GHZ$_4$-symmetric states resides in the tetrahedron. Among nine
SLOCC classes we have examined five SLOCC classes, which results in three
linear hierarchies $L_{abc_2} \subset L_{a_4} \subset L_{a_2b_2} \subset
G_{abcd}$, $L_{a_20_{3\oplus\bar{1}}} \subset G_{abcd}$, and
$L_{0_{3\oplus\bar{1}}0_{3\oplus\bar{1}}} \subset G_{abcd}$ which hold, at
least, in the whole set of the GHZ$_4$-symmetric states. Difficulties arising
in the analysis of the remaining SLOCC classes are briefly discussed.
