Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C5×Dic7 |
| derived, Sylow | C7 ⋊ C2×C3×C4×C5 |
| Hall | C3×C5×C7 ⋊ C2×C4 |
| Hall | C5×C7 ⋊ C2×C3×C4 |
| Hall | C3×C7 ⋊ C2×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C5 | D7 |
| commutator, hypocenter | C7 | C2×C3×C4×C5 |
| Frattini | C2 | C2×C3×C5×D7 |
| Fitting, socle, prime radical | C22×C3×C5×C7 | C2 |
| cosocle | C2×C7 | C22×C3×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3×C5 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 840 |
300 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×Dic5×Dic7, C3×#10587855, C3×#10587856, … |
| maximal normal subgroup of | C3×Dic5×Dic7, C3×#10587855, C3×#10587856, … |
| maximal quotient of | C2×C3×C5×C7⋊C8, C3×C5×#1136, C3×C4×C5×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 99);