Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of D7×ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×D7×ΓL(1,𝔽16) |
| unambiguous semidirect product | D5⋊(Dic3×D7) |
| unambiguous semidirect product | (D5×D7)⋊Dic3 |
| Fitting, socle | C2×C3×C5×C7 ⋊ C2×C4 |
| derived, Hall | C3×C5×C7 ⋊ C22×C4 |
| Hall | C5×C7 ⋊ C22×Dic3 |
| Hall | C3×C7 ⋊ C22×AGL(1,𝔽5) |
| Hall | C3×C5 ⋊ C2×C4×D7 |
| Sylow | C7 ⋊ C22×ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C2×Dic3×D7 |
| Sylow | C3 ⋊ C2×D7×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5×C7 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7 | C2×C4 |
| prime radical | C2×C3×D5×D7 | C2 |
| cosocle | C3×C7×D5 | C23 |
| LCS | G ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 | |
216 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 210 |
90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 961);