Finite Group DatabasebetaFinite Group Databasebeta| order | 6069 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C289 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×C289 |
| unambiguous semidirect product | C7⋊(C3×C289) |
| unambiguous semidirect product | (C7×C289)⋊C3 |
| Fitting, Hall | C7×C289 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C289 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C289 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C3×C289 |
| Frattini | C17 | C17×ΓL(1,𝔽8) |
| Fitting | C7×C289 | C3 |
| socle | C7×C17 | C3×C17 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C289 | |
12 conjugacy classes of subgroups.
1445 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6069, 2);