Finite Group DatabasebetaFinite Group Databasebeta| order | 8358 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D199 |
| Fitting, socle, derived, Hall | C7×C199 ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×C199 ⋊ C2 |
| Hall | D1393 ⋊ C3 |
| Sylow | C199 ⋊ AGL(1,𝔽7) |
| Hall | ΓL(1,𝔽8) ⋊ D199 |
| Sylow | C7 ⋊ C3×D199 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C199 | C2×C3 |
| LCS | G ▷ C7×C199 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,7,199|c^a=c^6,d^a=d^198,c^b=c^2502 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8358, 22);