Get ready Part 23

Let $S$ be the set of ordered triples $(a,b,c)$ of distinct elements of a finite set $A$ . Suppose that

1. $(a,b,c) \in S$ if and only if $(b,c,a) \in S$ ;

2. $(a,b,c) \in S$ if and only if $(c,b,a) \notin S$ ;

3. $(a,b,c)$ and $(c,d,a)$ are both in S if and only if $(b,c,d)$ and $(d,a,b)$ are both in $S$ .

Is there exists a one-to-one function $g$ from $A$ to $\mathbb{R}$ such that $g(a)<g(b)<g(c)$ implies $(a,b,c) \in S$ ?