Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| derived, Hall | C7×C127 ⋊ C22×C9 |
| Hall | C7⋊C9×C127 ⋊ C22 |
| Hall | D7×D127 ⋊ C9 |
| Sylow | C127 ⋊ C2×D7⋊C9 |
| Hall | C7⋊C9 ⋊ C2×D127 |
| Sylow | C7 ⋊ C2×C9×D127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | AGL(1,𝔽7)×D127 |
| commutator, hypocenter | C7×C127 | C22×C9 |
| Fitting, socle, cosocle | C3×C7×C127 | C22×C3 |
| prime radical | C3×D7×D127 | C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C3 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 8001 |
1365 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 42);