Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#63306⋊C3 |
| unambiguous semidirect product | (C2×#63306)⋊C3 |
| Hall | C2×#63306 ⋊ C3 |
| Hall | C5×ΓL(1,𝔽8) ⋊ C2×D4 |
| Hall | C5×C7 ⋊ C2×C3×D4 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C22⋊D5 |
| Sylow | C7 ⋊ C2×C3×C22⋊D5 |
| Sylow | C5 ⋊ C2×C4⋊AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D5×AGL(1,𝔽7) |
| commutator, cosocle | C2×C5×C7 | C23×C3 |
| Frattini | C2 | C2×D5×AGL(1,𝔽7) |
| Fitting, socle | C22×C5×C7 | C22×C3 |
| hypocenter | C5×C7 | C2×C3×D4 |
| LCS | G ▷ C2×C5×C7 ▷ C5×C7 | |
| UCS | C1 ◁ C22 | |
216 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 280 |
94 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 452);