Finite Group DatabasebetaFinite Group Databasebeta| order | 9009 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C9×C11×C13 · Schur multiplier (H2) = C1 |
| direct product | C11×C13×C7⋊C9 |
| unambiguous semidirect product | C7⋊(C9×C11×C13) |
| unambiguous semidirect product | (C7×C11)⋊(C9×C13) |
| derived, Sylow | C7 ⋊ C9×C11×C13 |
| Hall | C7×C11×C13 ⋊ C9 |
| Hall | C7×C13 ⋊ C9×C11 |
| Hall | C7×C11 ⋊ C9×C13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C11×C13 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C9×C11×C13 |
| Frattini | C3 | C11×C13×ΓL(1,𝔽8) |
| Fitting, socle | C3×C7×C11×C13 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C11×C13 | |
24 conjugacy classes of subgroups.
2145 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C11×C13×C7⋊C9, C11×D13×C7⋊C9, C13×D11×C7⋊C9, … |
| maximal normal subgroup of | C2×C11×C13×C7⋊C9, C11×D13×C7⋊C9, C13×D11×C7⋊C9, … |
| maximal quotient of | C2×C11×C13×C7⋊C9, C11×C13×C22⋊(C7⋊C9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9009, 3);