If■is a group of order g=pg',(p a prime number,(g',p)=1),rather complete results can be ebtaiued describing the behaviour of thecharacters with regard to the prime number p.In this paper,groups oforder g=p~2g',((p,g')=1)are inverstigated.The situation here is farmore complicated,since not only blocks of defects 0 and 1 but also blocksof defect 2 appear about which not much is known.The group(?)contains a group μof older p~2.We have here twocares:μmay be abelian of type(p,p)or(?)may be cyclic.We shall re-strict our work to the first case.Actually the method apply to the secondcase too and the situation there is much simpler than in the first case.We shall consider the nomalizers and centralizers of(?)and of the sub-groups of order p of(?).Those groups may be considered as of a less com-plicated nature than(?),since we can describe them easily by means ofgroups whose orders contain the given prime p only to the first power.The aim of our inveratigation is to show that the behaviour of thecharacters of (?) depends strongly on the structure of these subgroups.Anumber of connections between the characters of (?) and the characters ofthe snbgroups mentioned are obtained.We begin with a discussion of the centralizer (?) and the norma-lizer (?).The group (?) is the direct product (?)×(?) of (?) and a group(?) of an order prime to p.The faotor group is either abelian or a groupof well known type.The blocks of defect 2 of (?) are in one-to-one cor- respondence to the classes of irreducible characters (?) of (?) which areassociated in (?).Every block of defect 1 has a defect group (?) of orderp which is determined uniquely,if coujugate groups are considered as notessontially different.The centralizer (?) is again a direct product(?)×(?) and the blocks with the defect group (?) are in one-to-one corre-spondence to the classes of characters of defect zero of (?) associated in thenormalizer (?)of (?).The values of the characters of (?) are described largely by the de-composition numbers d_(μ(?))~i.The d_(μ(?))~i are elements of a cyclotomic field.Inthe later sections,the nature of the occuring irrationallities is studiedmore closely.In particular,we are again interested in the question towhat extent the nature of the irrationalities depends on the structure ofthe groups (?),(?),(?)and(?).I an indebted to Professor Richard Brauer for his help.