Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D5×C22⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×D5×C22⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | C5⋊(C22×C22⋊ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C22⋊ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C7×D5)⋊A4 |
| Fitting, socle | C23×C5×C7 ⋊ C2×C3 |
| derived | C22×C5×C7 ⋊ C22×C3 |
| Hall | C23×C7×D5 ⋊ C3 |
| Hall | C23×D5 ⋊ ΓL(1,𝔽8) |
| Hall | C5×C7 ⋊ C22×A4 |
| Sylow | C7 ⋊ C2×D5×A4 |
| Sylow | C5 ⋊ C22×C22⋊ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C5×C7 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5×C7 | C2×C3 |
| LCS | G ▷ C22×C5×C7 | |
| UCS | C1 ◁ C2 | |
156 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 210 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 979);