Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C2×C13×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C13 |
| prime radical | C2×D4×C13 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C4×C13 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23);(3,4,5,6,7,8,9,10);(4,6)(5,9)(8,10)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 194);