Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of AGL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C2×C3×C5×AGL(1,𝔽8) |
| unambiguous semidirect product | (C24×C3×C5)⋊C7 |
| Fitting, socle, Hall | C24×C3×C5 ⋊ C7 |
| derived | C23 ⋊ C2×C3×C5×C7 |
| Hall | C24×C5 ⋊ C3×C7 |
| Hall | C24×C3 ⋊ C5×C7 |
| Sylow | C24 ⋊ C3×C5×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C2×C3×C5×C7 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C2×C3×C5 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 210 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×C5×AΓL(1,𝔽8) |
| maximal normal subgroup of | C2×C3×C5×AΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 967);