  The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to
study some extensions of $q$-groups by nilpotent groups and the power of these
groups in a certain computational model. In its simplest formulation, it
establishes exponential lower bounds for $\mathrm{AND}_d \circ \mathrm{MOD}_m
\circ \mathrm{MOD}_q$ circuits computing AND of unbounded arity $n$ (for
constant integers $d,m$ and a prime $q$). While it has been proved in some
special cases (including $d=1$), it remains wide open in its general form for
over 30 years.
  In this paper we prove that the hypothesis holds when we restrict our
attention to symmetric circuits with $m$ being a prime. While we build upon
techniques by Grolmusz and Tardos (2000), we have to prove a new symmetric
version of their Degree Decreasing Lemma and apply it in a highly non-trivial
way. Moreover, to establish the result we perform a careful analysis of
automorphism groups of $\mathrm{AND} \circ \mathrm{MOD}_m$ subcircuits and
study the periodic behaviour of the computed functions.
  Finally, our methods also yield lower bounds when $d$ is treated as a
function of $n$.
