Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×C5×D7 |
| Fitting, socle | C22×C3×C5×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C23×C3×C5 |
| Hall | C3×C5×C7 ⋊ C23 |
| Hall | C5×C7 ⋊ C23×C3 |
| Hall | C3×C7 ⋊ C23×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C5 | D7 |
| commutator, cosocle, hypocenter | C7 | C23×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C5×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3×C5 | |
128 conjugacy classes of subgroups.
| transitive degree | μt(G) = 420 |
300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 179);