Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C7×AGL(1,𝔽5) |
| unambiguous semidirect product | D5⋊(C3×C4×C7) |
| Fitting, socle | C2×C3×C5×C7 ⋊ C4 |
| derived, Sylow | C5 ⋊ C2×C3×C4×C7 |
| Hall | C3×C5×C7 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C2×C3×C4 |
| Hall | C3×C5 ⋊ C2×C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C2×C3×C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7 | C4 |
| prime radical | C2×C3×C7×D5 | C2 |
| cosocle | D5 | C22×C3×C7 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C3×C7 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 210 |
210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 165);