Finite Group DatabasebetaFinite Group Databasebeta| order | 9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C17⋊(ΓL(1,𝔽8)⋊D7) |
| unambiguous semidirect product | (C2×C17)⋊(ΓL(1,𝔽8)⋊D7) |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊D119 |
| Fitting, socle | C2×C72×C17 ⋊ C2×C3 |
| derived, Hall | C72×C17 ⋊ C22×C3 |
| Hall | C2×C7⋊D119 ⋊ C3 |
| Hall | C7×C17×ΓL(1,𝔽8) ⋊ C22 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×D17 |
| Sylow | C72 ⋊ C2×C3×D17 |
| Sylow | C17 ⋊ C2×ΓL(1,𝔽8)⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C17⋊(ΓL(1,𝔽8)⋊D7) |
| commutator, cosocle, hypocenter | C72×C17 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C72×C17 | C2×C3 |
| LCS | G ▷ C72×C17 | |
| UCS | C1 ◁ C2 | |
100 conjugacy classes of subgroups.
604 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(9996, 126);