Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C17 · Schur multiplier (H2) = C2 |
| direct product | C2×C17×ΓL(1,𝔽8)⋊D7 |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊(D7×C17) |
| Fitting, socle | C2×C72×C17 ⋊ C2×C3 |
| derived, Sylow | C72 ⋊ C22×C3×C17 |
| Hall | C2×C17×C7⋊D7 ⋊ C3 |
| Hall | C7×C17×ΓL(1,𝔽8) ⋊ C22 |
| Hall | C72×C17 ⋊ C22×C3 |
| Hall | C2×C7⋊D7 ⋊ C3×C17 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C22×C17 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C17 | ΓL(1,𝔽8)⋊D7 |
| commutator, cosocle, hypocenter | C72 | C22×C3×C17 |
| Frattini | C1 | G |
| Fitting, socle | C2×C72×C17 | C2×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C2×C17 | |
100 conjugacy classes of subgroups.
748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 125);