Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D7×D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×Dic7×D15 |
| derived, Hall | C3×C5×C7 ⋊ C22×C4 |
| Hall | C5×C7 ⋊ C2×C4×D3 |
| Hall | C3×C7 ⋊ C2×C4×D5 |
| Hall | C3×C5 ⋊ C22×Dic7 |
| Sylow | C7 ⋊ C2×C4×D15 |
| Sylow | C5 ⋊ C2×D3×Dic7 |
| Sylow | C3 ⋊ C2×D5×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7×D15 |
| commutator, hypocenter | C3×C5×C7 | C22×C4 |
| Frattini | C2 | C2×D7×D15 |
| Fitting, socle | C22×C3×C5×C7 | C22 |
| prime radical | C22×C7×D15 | C2 |
| cosocle | C2×C3×C5×C7 | C23 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C22 | |
216 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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, 722);