Finite Group DatabasebetaFinite Group Databasebeta| order | 840 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C5×C7×SL(2,𝔽3) |
| unambiguous semidirect product | Q8⋊(C3×C5×C7) |
| unambiguous semidirect product | (C5×Q8)⋊(C3×C7) |
| Fitting, Hall | C5×C7×Q8 ⋊ C3 |
| derived, Sylow | Q8 ⋊ C3×C5×C7 |
| Hall | C7×Q8 ⋊ C3×C5 |
| Hall | C5×Q8 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C5×C7 | A4 |
| commutator, cosocle, hypocenter | Q8 | C3×C5×C7 |
| Frattini | C2 | C5×C7×A4 |
| Fitting | C5×C7×Q8 | C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C5×C7 | |
28 conjugacy classes of subgroups.
| transitive degree | μt(G) = 280 |
245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(840, 94);