Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C127 · Schur multiplier (H2) = C2×C3 |
| direct product | A4×ΓL(1,𝔽8)×C127 |
| unambiguous semidirect product | (C22×ΓL(1,𝔽8))⋊(C3×C127) |
| unambiguous semidirect product | (C7×A4)⋊(C3×C127) |
| unambiguous semidirect product | (C22⋊ΓL(1,𝔽8))⋊(C3×C127) |
| Fitting, socle, Hall | C22×C7×C127 ⋊ C32 |
| derived, Hall | C22×C7 ⋊ C32×C127 |
| Hall | C7×C127 ⋊ C3×A4 |
| Hall | C22×C127 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×A4×C127 |
| Sylow | C22 ⋊ C3×ΓL(1,𝔽8)×C127 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C127 | A4×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C7 | C32×C127 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7×C127 | C32 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C127 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 3556 |
2540 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 225);