Section ExtremalClass.
Variables (gT : finGroupType) (G : {group gT}).
Inductive extremal_group_type :=
ModularGroup | Dihedral | SemiDihedral | Quaternion | NotExtremal.
Definition index_extremal_group_type c :=
match c with
| ModularGroup => 0
| Dihedral => 1
| SemiDihedral => 2
| Quaternion => 3
| NotExtremal => 4
end%N.
Definition enum_extremal_groups :=
[:: ModularGroup; Dihedral; SemiDihedral; Quaternion].
End ExtremalClass.

(* This is Aschbacher (23.4).  *)
Lemma maximal_cycle_extremal gT p (G X : {group gT}) :
p.-group G -> ~~ abelian G -> cyclic X -> X \subset G -> #|G : X| = p ->
(extremal_class G == ModularGroup) || (p == 2) && extremal2 G.