Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C9×C16 · Schur multiplier (H2) = C1 |
| direct product | C16×C7⋊C9 |
| unambiguous semidirect product | (C7×C16)⋊C9 |
| derived, Sylow | C7 ⋊ C9×C16 |
| Hall | C7×C16 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C16 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C9×C16 |
| Frattini | C3×C8 | C2×ΓL(1,𝔽8) |
| Fitting | C3×C7×C16 | C3 |
| socle, prime radical | C2×C3×C7 | C3×C8 |
| cosocle | C3×C7×C8 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C16 | |
30 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1008 |
240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 2);