Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)×D381 |
| unambiguous semidirect product | (C2×C3)⋊(ΓL(1,𝔽8)×D127) |
| Fitting, socle | C2×C3×C7×C127 ⋊ C2×C3 |
| derived | C3×C7×C127 ⋊ C22×C3 |
| Hall | C3×ΓL(1,𝔽8)×C127 ⋊ C22 |
| Hall | C7×C127 ⋊ C2×C3×D3 |
| Sylow | C127 ⋊ C2×D3×ΓL(1,𝔽8) |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×D127 |
| Sylow | C7 ⋊ C2×C3×D381 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×D381 |
| commutator, cosocle, hypocenter | C3×C7×C127 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7×C127 | C2×C3 |
| LCS | G ▷ C3×C7×C127 | |
| UCS | C1 ◁ C2 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 5334 |
1920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 261);