Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | ΓL(1,𝔽8)×D20 |
| unambiguous semidirect product | C4⋊(D5×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C7×D20)⋊C3 |
| Fitting | C4×C5×C7 ⋊ C2×C3 |
| Hall | C7×D20 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ D4 |
| Hall | C5×C7 ⋊ C3×D4 |
| Sylow | C7 ⋊ C3×D20 |
| Sylow | C5 ⋊ D4×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C2×C5×C7 | C22×C3 |
| Fitting | C4×C5×C7 | C2×C3 |
| hypercenter | C4 | D5×ΓL(1,𝔽8) |
| hypocenter | C5×C7 | C3×D4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 140 |
65 conjugacy classes — view the full character table.
| maximal subgroup of | #10587841⋊C3, #10587843⋊C3, #10587845⋊C3, … |
| maximal normal subgroup of | #10587841⋊C3, #10587843⋊C3, #10587845⋊C3, … |
| maximal quotient of | ΓL(1,𝔽8)×#677, ΓL(1,𝔽8)×D40, ΓL(1,𝔽8)×Dic20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 30);