Finite Group DatabasebetaFinite Group Databasebeta| order | 7308 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)×D87 |
| unambiguous semidirect product | (C2×C3)⋊(ΓL(1,𝔽8)×D29) |
| unambiguous semidirect product | (C2×C29)⋊(D3×ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C2×C7×D87)⋊C3 |
| Fitting, socle | C2×C3×C7×C29 ⋊ C2×C3 |
| derived | C3×C7×C29 ⋊ C22×C3 |
| Hall | C3×ΓL(1,𝔽8)×C29 ⋊ C22 |
| Hall | C7×C29 ⋊ C2×C3×D3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×D29 |
| Sylow | C29 ⋊ C2×D3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×C3×D87 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×D87 |
| commutator, hypocenter | C3×C7×C29 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7×C29 | C2×C3 |
| LCS | G ▷ C3×C7×C29 | |
| UCS | C1 ◁ C2 | |
88 conjugacy classes of subgroups.
450 conjugacy classes — view the full character table.
| maximal subgroup of | D5×ΓL(1,𝔽8)×D87, C2×C5×ΓL(1,𝔽8)×D87, C2×ΓL(1,𝔽8)×D435 |
| maximal normal subgroup of | C2×C5×ΓL(1,𝔽8)×D87 |
| maximal quotient of | D5×ΓL(1,𝔽8)×D87, C2×C5×ΓL(1,𝔽8)×D87, C2×ΓL(1,𝔽8)×D435 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7308, 116);