Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of A4×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×A4×Dic7 |
| derived | C22×C7 ⋊ C3×C4×C5 |
| Hall | C22×C5×Dic7 ⋊ C3 |
| Hall | C22×Dic7 ⋊ C3×C5 |
| Hall | C5×C7 ⋊ C4×A4 |
| Sylow | C7 ⋊ C4×C5×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | A4×D7 |
| commutator, hypocenter | C22×C7 | C3×C4×C5 |
| Frattini | C2 | C5×A4×D7 |
| Fitting, socle | C23×C5×C7 | C2×C3 |
| prime radical | C2×C5×C7×A4 | C2 |
| cosocle | C23×C7 | C2×C3×C5 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C2×C5 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 420 |
200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 607);