In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset.
That ordering, $\prec$, was defined in section 4.6 as follows.
Throughout this section, we shall assume that a $BSGS (B, S)$ with $B= \left[\beta_1,\ldots , \beta_k\right]$ is known for the permutation group $G$, and we shall use all of the notation that was introduced in Subsection 4.4.1. For $0\le l \le k$, we shall denote the initial segment $\left[\beta_1,\ldots, \beta_l\right]$ of $B$ by $B(l)$.
It is convenient to introduce an ordering on $\Omega = \left[1 \ldots n\right]$ in which the base elements come first, and in order. That is, $\beta_i \prec \beta_j$, for $i<j$, and $\beta_i \prec \alpha$ if $\alpha \notin Β$. By the definition of a base, an element $g\in G$ is uniquely determined by its base image $B^g = \left[\beta^g_1, \ldots, \beta^g_k\right]$, and so we can order the elements of $G$ by their base images. That is, for $g$, $h\in G$, we define $g \prec h$ if $B^g$ precedes $B^h$ in the lexicographical ordering induced by $≺$. To be precise $g \prec h$, if and only if, for some $l$ with $1\le l \le k$, we have $\beta^g_i = \beta^h_i$ for $1\le i < l$ and $\beta^g_l \prec \beta^h_l$.
So, the concept is standing on the ordering of the elements in the base $B$.
My question:
Is the ordering in $B= \left[\beta_1,\ldots , \beta_k\right]$ arbitrary? If not, is it ordered lexicographically?