Fact gtype_key : unit. Proof. by []. Qed.
Definition gtype := locked_with gtype_key (sdprod_groupType gact).
Hypotheses (p_gt1 : p > 1) (q_gt1 : q > 1).

Lemma Grp : (exists s, [/\ s \in Aut B, #[s] %| p & s b = b ^+ e]) ->
[set: gtype] \isog Grp (x : y : (x ^+ q, y ^+ p, x ^ y = x ^+ e)).