Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D4×C5⋊AGL(1,𝔽7) |
| unambiguous semidirect product | (D4×D35)⋊C3 |
| Fitting | C5×C7×D4 ⋊ C2×C3 |
| Hall | D4×D35 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C2×D4 |
| Hall | C5×C7 ⋊ C2×C3×D4 |
| Hall | ΓL(1,𝔽8) ⋊ D4×D5 |
| Sylow | C7 ⋊ C3×D4×D5 |
| Sylow | C5 ⋊ D4×AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C5⋊AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C5×C7 | C23×C3 |
| Fitting | C5×C7×D4 | C2×C3 |
| hypercenter | D4 | C5⋊AGL(1,𝔽7) |
| hypocenter | C5×C7 | C2×C3×D4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C2 ◁ D4 | |
216 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
85 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 599);