Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C22×C3×C4×D4 (order 384) · Inn(G) = D4 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | D4 |
| commutator | C4 | C2×C4×C13 |
| Frattini, cosocle | C2×C4 | C22×C13 |
| socle | C2×C13 | C22⋊C4 |
| prime radical | C13×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4,7,5,3,8,6)(9,10,11,12,13,14,15,16,17,18,19,20,21);(1,3)(2,5)(4,6)(7,8)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 54);