Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of C7⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×C7⋊S4 |
| unambiguous semidirect product | (C2×C7)⋊(C5×S4) |
| unambiguous semidirect product | (C2×A4)⋊(C5×D7) |
| Fitting, socle | C23×C5×C7 ⋊ D3 |
| derived | C7×A4 ⋊ C22×C5 |
| Hall | C5×C7 ⋊ C2×S4 |
| Sylow | C7 ⋊ C2×C5×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | C7⋊S4 |
| commutator, cosocle, hypocenter | C7×A4 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5×C7 | D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C2×C5 | |
132 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 210 |
170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 969);