  The catenary data of a matroid $M$ of rank $r$ on $n$ elements is the vector
$(\nu(M;a_0,a_1,\ldots,a_r))$, indexed by compositions $(a_0,a_1,\ldots,a_r)$,
where $a_0 \geq 0$,\, $a_i > 0$ for $i \geq 1$, and $a_0+ a_1 + \cdots + a_r =
n$, with the coordinate $\nu (M;a_0,a_1, \ldots,a_r)$ equal to the number of
maximal chains or flags $(X_0,X_1, \ldots,X_r)$ of flats or closed sets such
that $X_i$ has rank $i$,\, $|X_0| = a_0$, and $|X_i - X_{i-1}| = a_i$. We show
that the catenary data of $M$ contains the same information about $M$ as its
$\mathcal{G}$-invariant, which was defined by H. Derksen [\emph{J.\ Algebr.\
Combin.}\ 30 (2009) 43--86]. The Tutte polynomial is a specialization of the
$\mathcal{G}$-invariant. We show that many known results for the Tutte
polynomial have analogs for the $\mathcal{G}$-invariant. In particular, we show
that for many matroid constructions, the $\mathcal{G}$-invariant of the
construction can be calculated from the $\mathcal{G}$-invariants of the
constituents and that the $\mathcal{G}$-invariant of a matroid can be
calculated from its size, the isomorphism class of the lattice of cyclic flats
with lattice elements labeled by the rank and size of the underlying set. We
also show that the number of flats and cyclic flats of a given rank and size
can be derived from the $\mathcal{G}$-invariant, that the
$\mathcal{G}$-invariant of $M$ is reconstructible from the deck of
$\mathcal{G}$-invariants of restrictions of $M$ to its copoints, and that,
apart from free extensions and coextensions, one can detect whether a matroid
is a free product from its $\mathcal{G}$-invariant.
