Function automorphism_group_order 

pub fn automorphism_group_order(
    degree: usize,
    gens: &[Perm],
    cap: usize,
) -> Option<u128>

The order of the automorphism group Aut(G) of G = ⟨gens⟩ — the symmetries of the group itself (bijections G → G preserving multiplication, φ(xy) = φ(x)φ(y)). An automorphism is determined by the images of a generating set, so the search ranges over candidate images (each generator must map to an element of the same order, a necessary condition) and accepts those that extend to a consistent, bijective homomorphism. None when |G| > cap or the candidate search would exceed its budget. Classic: |Aut(Cₙ)| = φ(n), |Aut(Sₙ)| = n! (n≠6), |Aut(V₄)| = 6, |Aut(D₄)| = 8, |Aut(Q₈)| = 24.