// The binary dihedral group BD48.
// It has 4 1-dim and 11 2-dimensional representations.
K<ep> := CyclotomicField(24);
GL := GeneralLinearGroup(2,K);
A := elt< GL | ep, 0, 0, ep^-1 >;
B := elt< GL | 0,1,-1,0 >;
// I check that A and B satisfy the relations of BD48.
Order(A); Order(B); A^12 eq B^2; A*B*A eq B;
BD := sub< GL | [A,B] >; // matrix group gen by A and B
Order(BD);
BD;
ConjugacyClasses(BD); // There are 14 conjugacy classes.
// two of them have 12 elements. Ask for all g in BD conjugate
// to specimen elt of ConjugacyClass(BD)[5].
[g : g in BD | IsConjugate(BD, g, $1[5,3])];
CharacterTable(BD);