  The problem of finding the number of ordered commuting tuples of elements in
a finite group is equivalent to finding the size of the solution set of the
system of equations determined by the commutator relations that impose
commutativity among any pair of elements from an ordered tuple. We consider
this type of systems for the case of ordered triples and express the size of
the solution set in terms of the irreducible characters of the group. The
obtained formulas are natural extensions of Frobenius' character formula that
calculates the number of ways a group element is a commutator of an ordered
pair of elements in a finite group. We discuss how our formulas can be used to
study the probability distributions afforded by these systems of equations, and
we show explicit calculations for dihedral groups.
