  In this work we show the step by step calculations needed to quantify the
contribution of a three-loop order diagram with dihedral symmetry to the
radiative corrections of the pressure in SU(2) thermal Yang-Mills theory in
deconfining phase. We surveyed past developments, and performed computations
for separate channel combinations, defined by Mandelstam variables which are
constrained by two 4-vertices. An analytically integrable approximation for
high-temperature conditions was found, to verify the relevance of the
corrections for this diagram. A numerical analysis with Monte Carlo methods was
carried out to check the validity of such approximation, to compare it with the
full integral. A Dyson-Schwinger resummation had to be performed to all
dihedral loop orders in order to control the temperature dependency found.
