Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of C22⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C22×C5×C22⋊ΓL(1,𝔽8) |
| Fitting, socle, Hall | C24×C5×C7 ⋊ C3 |
| derived | C22×C7 ⋊ C22×C3×C5 |
| Hall | C24×C7 ⋊ C3×C5 |
| Hall | C24×C5 ⋊ ΓL(1,𝔽8) |
| Hall | C5×C7 ⋊ C22×A4 |
| Sylow | C24 ⋊ C5×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C22×C5×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C7 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C5×C7 | C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C22×C5 | |
156 conjugacy classes of subgroups.
| transitive degree | μt(G) = 420 |
240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 995);