Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C22×ΓL(1,𝔽8)×C127⋊C3 |
| Fitting, socle, Hall | C22×C7×C127 ⋊ C32 |
| derived, Hall | C7×C127 ⋊ C22×C32 |
| Hall | C22×C127 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C127 ⋊ C22×C3×ΓL(1,𝔽8) |
| Hall | C22×C7 ⋊ C3×C127⋊C3 |
| Sylow | C7 ⋊ C22×C3×C127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ΓL(1,𝔽8)×C127⋊C3 |
| commutator, cosocle, hypocenter | C7×C127 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7×C127 | C32 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C22 | |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 3556 |
900 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 234);