Section Extraspecial.
Variables (F : fieldType) (gT : finGroupType) (S : {group gT}) (p n : nat).
Hypotheses (pS : p.-group S) (esS : extraspecial S).
Hypothesis oSpn : #|S| = (p ^ n.*2.+1)%N.
Hypotheses (splitF : group_splitting_field F S) (F'S : [char F]^'.-group S).
Let p_pr := extraspecial_prime pS esS.
Let p_gt0 := prime_gt0 p_pr.
Let p_gt1 := prime_gt1 p_pr.
Let oZp := card_center_extraspecial pS esS.
Let modIp' (i : 'I_p.-1) : (i.+1 %% p = i.+1)%N.
(* This is the corolloray of the above that is actually used in the proof of  *)
(* B & G, Theorem 2.5. It encapsulates the dependency on a socle of the       *)
(* regular representation.                                                    *)
Variables (m : nat) (rS : mx_representation F S m) (U : 'M[F]_m).
Hypotheses (simU :  mxsimple rS U) (ffulU : rstab rS U == 1%g).
Let sZS := center_sub S.
Let rZ := subg_repr rS sZS.

Lemma faithful_repr_extraspecial :
\rank U = (p ^ n)%N /\
(forall V, mxsimple rS V -> mx_iso rZ U V -> mx_iso rS U V).