Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of D5×ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×#168 |
| unambiguous semidirect product | (C7×#168)⋊C3 |
| derived, Hall | C5×C7 ⋊ C3×C8 |
| Hall | C7×#168 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C8 |
| Sylow | C7 ⋊ C3×#168 |
| Sylow | C5 ⋊ C8×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D5×ΓL(1,𝔽8) |
| commutator, hypocenter | C5×C7 | C3×C8 |
| Fitting, cosocle | C4×C5×C7 | C2×C3 |
| socle | C2×C5×C7 | C3×C4 |
| prime radical | C2×C5×ΓL(1,𝔽8) | C4 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C4 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 2);