Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C127 · Schur multiplier (H2) = C3 |
| direct product | C7×#157×C127 |
| Fitting, socle, Hall | C32×C7×C127 ⋊ C4 |
| derived, Sylow | C32 ⋊ C4×C7×C127 |
| Hall | C32×C127 ⋊ C4×C7 |
| Hall | C32×C7 ⋊ C4×C127 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C127 | #157 |
| commutator, hypocenter | C32 | C4×C7×C127 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7×C127 | C4 |
| prime radical | C7×C3⋊D3×C127 | C2 |
| cosocle | C3⋊D3 | C2×C7×C127 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C7×C127 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 5334 |
5334 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 264);