Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
| unambiguous semidirect product | D5⋊(ΓL(1,𝔽8)⋊C4) |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊AGL(1,𝔽5) |
| unambiguous semidirect product | (C2×C7⋊AGL(1,𝔽5))⋊C3 |
| Fitting, socle | C2×C5×C7 ⋊ C3×C4 |
| derived, Hall | C5×C7 ⋊ C2×C3×C4 |
| Hall | C2×C7⋊AGL(1,𝔽5) ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C2×C4 |
| Hall | ΓL(1,𝔽8) ⋊ C2×AGL(1,𝔽5) |
| Sylow | C7 ⋊ C2×C3×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C2×ΓL(1,𝔽8)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
| commutator, hypocenter | C5×C7 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C5×C7 | C3×C4 |
| prime radical | C2×D5×ΓL(1,𝔽8) | C2 |
| cosocle | C7×D5 | C22×C3 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 70 |
38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 142);