Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C127 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊D21×C127 |
| unambiguous semidirect product | (C2×C3)⋊(D21×C127) |
| unambiguous semidirect product | (C2×C7)⋊(C3⋊D3×C127) |
| Fitting, socle | C2×C32×C7×C127 ⋊ C2 |
| derived, Hall | C32×C7 ⋊ C22×C127 |
| Hall | C32×C7×C127 ⋊ C22 |
| Hall | C32×C127 ⋊ C2×D7 |
| Hall | C7×C127 ⋊ C2×C3⋊D3 |
| Sylow | C32 ⋊ C2×D7×C127 |
| Sylow | C7 ⋊ C2×C3⋊D3×C127 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C127 | C3⋊D21 |
| commutator, cosocle, hypocenter | C32×C7 | C22×C127 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7×C127 | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2×C127 | |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 16002 |
8382 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 309);