ClassMultCoeffCharTable( tbl, c1, c2, c3 )
returns the class multiplication coefficient of the classes c1, c2 and c3 of the group G with character table tbl.
gap> t:= CharTable( "L3(2)" );;
gap> ClassMultCoeffCharTable( t, 2, 2, 4 );
4
The class multiplication coefficient c_{1,2,3} of the classes c1, c2 and c3 equals the number of pairs (x,y) of elements x, y in G such that x lies in class c1, y lies in class c2 and their product xy is a fixed element of class c3.
Also in the center of the group algebra these numbers are found as coefficients of the decomposition of the product of two class sums K_i and K_j into class sums, [ K_i K_j = sum_k c_ijk K_k. ]
Given the character table of a finite group G, whose classes are C_1, dots, C_r with representatives g_i in C_i, the class multiplication coefficients c_{ijk} can be computed by the following formula.
[c_ijk = frac|C_i||C_j||G| sum_chi in mboxfootnotesize Irr(G) fracchi(g_i) chi(g_j) overlinechi(g_k)chi(1) ]
On the other hand the knowledge of the class multiplication coefficients enables the computation of the character table (see CharTable).
GAP 3.4.4