Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of D3×D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | D3×C7×Dic5 |
| derived, Hall | C3×C5 ⋊ C2×C4×C7 |
| Hall | C3×C5×C7 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C4×D3 |
| Hall | C3×C7 ⋊ C2×Dic5 |
| Sylow | C5 ⋊ C4×D3×C7 |
| Sylow | C3 ⋊ C2×C7×Dic5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D3×D5 |
| commutator, hypocenter | C3×C5 | C2×C4×C7 |
| Frattini | C2 | D3×C7×D5 |
| Fitting, socle | C2×C3×C5×C7 | C22 |
| prime radical | C2×C5×D3×C7 | C2 |
| cosocle | C2×C3×C5 | C22×C7 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C7 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 420 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C7×#170, C7×#6739, D3×C7⋊#170, … |
| maximal normal subgroup of | D3×C7×#170, C7×#6739, D3×C7⋊#170, … |
| maximal quotient of | D3×C7×#168, C7×#6650, C7×Dic3×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 64);