Section CenterMode.
Variable i : sG.
Let i0 := Ordinal (irr_degree_gt0 i).
Definition irr_mode x := irr_repr i x i0 i0.
End CenterMode.

Lemma cycle_repr_structure x (sG : irrType G) :
G :=: <[x]> -> [char F]^'.-group G -> group_splitting_field G ->
exists2 w : F, #|G|.-primitive_root w &
exists iphi : 'I_#|G| -> sG,
[/\ bijective iphi,
#|sG| = #|G|,
forall i, irr_mode (iphi i) x = w ^+ i
& forall i, irr_repr (iphi i) x = (w ^+ i)%:M].