Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C5 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×C5×D7 |
| Fitting, socle | C23×C3×C5×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C24×C3×C5 |
| Hall | C3×C5×C7 ⋊ C24 |
| Hall | C5×C7 ⋊ C24×C3 |
| Hall | C3×C7 ⋊ C24×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C5 | D7 |
| commutator, cosocle, hypocenter | C7 | C24×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C5×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C23×C3×C5 | |
536 conjugacy classes of subgroups.
| transitive degree | μt(G) = 840 |
600 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 998);