II, III, IV and V depends on the structure and embedding of the subgroup U as follows: Type II: U is nontrivial abelian and its normalizer is not contained in M. Type III: U is nontrivial abelian and its normalizer is contained in M. Type IV: U is nonabelian. Type V: U is trivial. All but two classes of maximal subgroups are of type I, but there may also be two extra classes of maximal subgroups, one of type II, and one of type II, III, IV, or V. Step 2. Character theory of G If X is an irreducible character of the normalizer H of the maximal abelian subgroup A of the CA group G, not containing A in its kernel, we can induce X to a character Y of G, which is not necessarily irreducible. Because of the known structure of G, it is easy to find the character values of Y on all but the identity element of G. This implies that if X1 and X2 are two such irreducible characters of H and Y1 and Y2 are the corresponding induced characters, then Y1 − Y2 is completely determined, and calculating its norm shows that it is the difference of two irreducible characters of G (these are sometimes known as exceptional characters of G with respect to H). A counting argument shows that each non-trivial irreducible character of G arises exactly once as an exceptional character associated to the normalizer of some maximal abelian subgroup of G. A similar argument (but replacing abelian Hall subgroups by nilpotent Hall subgroups) works in the proof of the CN-theorem. However, in the proof of the odd-order theorem, the arguments for constructing characters of G from characters of subgroups are far more delicate, and use the Dade isometry between character rings rather than character induction, since the maximal subgroups have a more complicated structure and are embedded in a less transparent way. The theory of exceptional characters is replaced by the theory of a coherent set of characters to extend the Dade isometry. Roughly speaking, this theory says that the Dade isometry can be extended unless the groups involved have a certain precise structure. described a simplified version the character theory due to Dade, Sibley, and Peterfalvi. Step 3. The final contradiction By step 2, we have a complete and precise description of the character table of the CA group G.