  The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been
extensively studied in complexity theory. For example, symmetric functions,
that is, functions that are invariant under the action of $S_n$, is an
important class of functions in the study of Boolean functions. A function
$f:\{0,1\}^n \to \{0,1\}$ is called transitive (or weakly-symmetric) if there
exists a transitive group $G$ of $S_n$ such that $f$ is invariant under the
action of $G$ - that is the function value remains unchanged even after the
bits of the input of $f$ are moved around according to some permutation $\sigma
\in G$. Understanding various complexity measures of transitive functions has
been a rich area of research for the past few decades.
  In this work, we study transitive functions in light of several combinatorial
measures. We look at the maximum separation between various pairs of measures
for transitive functions. Such study for general Boolean functions has been
going on for past many years. The best-known results for general Boolean
functions have been nicely compiled by Aaronson et. al (STOC, 2021).
  The separation between a pair of combinatorial measures is shown by
constructing interesting functions that demonstrate the separation. But many of
the celebrated separation results are via the construction of functions (like
"pointer functions" from Ambainis et al. (JACM, 2017) and "cheat-sheet
functions" Aaronson et al. (STOC, 2016)) that are not transitive. Hence, we
don't have such separation between the pairs of measures for transitive
functions.
  In this paper we show how to modify some of these functions to construct
transitive functions that demonstrate similar separations between pairs of
combinatorial measures.
