Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of A4×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×A4×D7 |
| Fitting, socle | C22×C5×C7 ⋊ C2×C3 |
| derived | C22×C7 ⋊ C2×C3×C5 |
| Hall | C22×C5×D7 ⋊ C3 |
| Hall | C22×D7 ⋊ C3×C5 |
| Hall | C5×C7 ⋊ C2×A4 |
| Sylow | C7 ⋊ C2×C5×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | A4×D7 |
| commutator, cosocle, hypocenter | C22×C7 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C5×C7 | C2×C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C5 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 140 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D7×S4, D7×C5⋊S4, D5×A4×D7, … |
| maximal normal subgroup of | C5×D7×S4, D7×C5⋊S4, D5×A4×D7, … |
| maximal quotient of | C5×#66662, C5×D7×SL(2,𝔽3), C5×A4×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 159);