Finite Group DatabasebetaFinite Group Databasebeta| order | 10689 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C509 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×C509 |
| unambiguous semidirect product | C7⋊(C3×C509) |
| unambiguous semidirect product | (C7×C509)⋊C3 |
| Fitting, socle, Hall | C7×C509 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C509 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C509 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C509 |
| Frattini | C1 | G |
| Fitting, socle | C7×C509 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C509 | |
8 conjugacy classes of subgroups.
2545 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10689, 1);