  Let H and K be finite composition series of a group G. The intersections
H_i\cap K_j of their members form a lattice CSL(H,\K) under set inclusion.
Improving the Jordan-H\"older theorem, G. Gr\"atzer, J.B. Nation and the
present authors have recently shown that H and K determine a unique permutation
pi such that, for all i, the i-th factor of H is "down-and-up projective" to
the pi(i)-th factor of K. Equivalent definitions of pi were earlier given by
R.P. Stanley and H. Abels. We prove that pi determines the lattice CLS(H,K).
More generally, we describe slim semimodular lattices, up to isomorphism, by
permutations, up to an equivalence relation called "sectionally inverted or
equal". As a consequence, we prove that the abstract class of all CSL(H,K)
coincides with the class of duals of all slim semimodular lattices.
