  Recently, Hyatt introduced some colored Eulerian quasisymmetric function to
study the joint distribution of excedance number and major index on colored
permutation groups. We show how Hyatt's generating function formula for the
fixed point colored Eulerian quasisymmetric functions can be deduced from the
Decrease value theorem of Foata and Han. Using this generating function
formula, we prove two symmetric function generalizations of the
Chung-Graham-Knuth symmetrical Eulerian identity for some flag Eulerian
quasisymmetric functions, which are specialized to the flag excedance numbers.
Combinatorial proofs of those symmetrical identities are also constructed.
  We also study some other properties of the flag Eulerian quasisymmetric
functions. In particular, we confirm a recent conjecture of Mongelli [Journal
of Combinatorial Theory, Series A, 120 (2013) 1216--1234] about the unimodality
of the generating function of the flag excedances over the type B derangements.
Moreover, colored versions of the hook factorization and admissible inversions
of permutations are found, as well as a new recurrence formula for the
$(\maj-\exc,\fexc)$-$q$-Eulerian polynomials.
  We introduce a colored analog of Rawlings major index on colored permutations
and obtain an interpretation of the colored Eulerian quasisymmetric functions
as sums of some fundamental quasisymmetric functions related with them, by
applying Stanley's $P$-partition theory and a decomposition of the Chromatic
quasisymmetric functions due to Shareshian and Wachs.
