Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of D3×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×Dic3×D7 |
| unambiguous semidirect product | (C3×D7)⋊(C4×C5) |
| derived, Hall | C3×C7 ⋊ C2×C4×C5 |
| Hall | C3×C5×C7 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C2×Dic3 |
| Hall | C3×C5 ⋊ C4×D7 |
| Sylow | C7 ⋊ C2×C5×Dic3 |
| Sylow | C3 ⋊ C4×C5×D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3×D7 |
| commutator, hypocenter | C3×C7 | C2×C4×C5 |
| Frattini | C2 | C5×D3×D7 |
| Fitting, socle | C2×C3×C5×C7 | C22 |
| prime radical | C2×C3×C5×D7 | C2 |
| cosocle | C2×C3×C7 | C22×C5 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C5 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 420 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic3×D7, D7×#1200, D7×#1202, … |
| maximal normal subgroup of | D5×Dic3×D7, D7×#1200, D7×#1202, … |
| maximal quotient of | C5×D7×C3⋊C8, C5×#66557, C5×Dic3×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 49);