Finite Group DatabasebetaFinite Group Databasebeta| order | 684 |
| isoclinism | isoclinic class of AGL(1,𝔽19) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×AGL(1,𝔽19) |
| unambiguous semidirect product | #4296⋊C22 |
| unambiguous semidirect product | (C2×#4296)⋊C2 |
| Fitting, socle | C2×C19 ⋊ C2×C9 |
| derived, Sylow | C19 ⋊ C22×C9 |
| Hall | #4296 ⋊ C22 |
| Hall | C2×D19 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,𝔽19) |
| commutator, hypocenter | C19 | C22×C9 |
| Frattini | C1 | G |
| Fitting, socle | C2×C19 | C2×C9 |
| prime radical | C2×D19⋊C3 | C3 |
| cosocle | C19⋊C3 | C22×C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2 | |
30 conjugacy classes of subgroups.
| transitive degree | μt(G) = 38 |
38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(684, 28);