Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D7×D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | D7×C3⋊#168 |
| unambiguous semidirect product | (C3×D7)⋊#168 |
| derived, Hall | C3×C5×C7 ⋊ C2×C8 |
| Hall | C5×C7 ⋊ C2×C3⋊C8 |
| Hall | C3×C7 ⋊ C2×#168 |
| Hall | C3×C5 ⋊ C8×D7 |
| Sylow | C7 ⋊ C2×C3⋊#168 |
| Sylow | C5 ⋊ D7×C3⋊C8 |
| Sylow | C3 ⋊ D7×#168 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D7×D15 |
| commutator, hypocenter | C3×C5×C7 | C2×C8 |
| Fitting, cosocle | C3×C4×C5×C7 | C22 |
| socle | C2×C3×C5×C7 | C2×C4 |
| prime radical | C2×C3×C5×D7 | C4 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C4 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 840 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 142);