Two positive definite symmetric
n × n matrices A, B with integer elements and determinant one are said to be
congruent if there exists an integral C such that B = CACT (CT is the
transpose of C). This is an equivalence relation. The number of equivalence
classes, C-classes, is finite and is known for all n ≦ 16. Let G be a finite
group of order n and let Y , Z be two positive definite symmetric group
matrices for G with integral elements and determinant one. If an integral
group matrix X for G exists such that Z = XY XT then Z, Y are said to be
G-congruent. G congruence is an equivalence relation. In this paper the
interlinking of the G-classes with the C-classes is determined for all groups of
order n ≦ 13. The principal result is that the G-class number is two for
certain groups of orders eight or twelve and is one for all other groups of order
n ≦ 13.
