Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×#11765919⋊D7 |
| unambiguous semidirect product | (C2×#11765919)⋊D7 |
| Fitting, socle | C2×C7×C127 ⋊ C2×C9 |
| derived, Hall | C7×C127 ⋊ C22×C9 |
| Hall | C7×#11765919 ⋊ C22 |
| Hall | C2×D889 ⋊ C9 |
| Hall | #11765919 ⋊ C2×D7 |
| Sylow | C127 ⋊ C2×C9×D7 |
| Sylow | C7 ⋊ C2×#11765919⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11765919⋊D7 |
| commutator, hypocenter | C7×C127 | C22×C9 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C127 | C2×C9 |
| prime radical | C2×(C127⋊C3)⋊D7 | C3 |
| cosocle | C7×C127⋊C3 | C22×C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1778 |
188 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 185);