Finite Group DatabasebetaFinite Group Databasebeta| order | 30324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)⋊D361 |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊D361 |
| Fitting | C2×C7×C361 ⋊ C2×C3 |
| derived, Hall | C7×C361 ⋊ C22×C3 |
| Hall | C2×D2527 ⋊ C3 |
| Hall | ΓL(1,𝔽8)×C361 ⋊ C22 |
| Sylow | C361 ⋊ C2×AGL(1,𝔽7) |
| Hall | ΓL(1,𝔽8) ⋊ C2×D361 |
| Sylow | C7 ⋊ C2×C3×D361 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)⋊D361 |
| commutator, cosocle, hypocenter | C7×C361 | C22×C3 |
| Frattini | C19 | C2×ΓL(1,𝔽8)⋊D19 |
| Fitting | C2×C7×C361 | C2×C3 |
| socle | C2×C7×C19 | C3×D19 |
| LCS | G ▷ C7×C361 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 5054 |
1814 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30324, 37);