Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×C3⋊#170 |
| unambiguous semidirect product | (C2×C3)⋊(C7×#170) |
| derived, Hall | C3×C5 ⋊ C2×C7×C8 |
| Hall | C3×C5×C7 ⋊ C2×C8 |
| Hall | C5×C7 ⋊ C2×C3⋊C8 |
| Hall | C3×C7 ⋊ C2×#170 |
| Sylow | C5 ⋊ C2×C7×C3⋊C8 |
| Sylow | C3 ⋊ C2×C7×#170 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C7 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C2×C7×C8 |
| Frattini | C2 | C2×C7×ΓL(1,𝔽16) |
| Fitting, socle, prime radical | C22×C3×C5×C7 | C4 |
| cosocle | C3×Dic5 | C22×C7 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C22×C7 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1680 |
252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 641);