Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C22⋊((C127⋊C3)⋊C7) |
| unambiguous semidirect product | (C22×C3)⋊((C127⋊C3)⋊C7) |
| unambiguous semidirect product | (C3×C127⋊C7)⋊A4 |
| unambiguous semidirect product | (C3×C22⋊(C127⋊C3))⋊C7 |
| Fitting, socle | C22×C3×C127 ⋊ C3×C7 |
| derived, Hall | C22×C127 ⋊ C32×C7 |
| Hall | C3×C22⋊(C127⋊C3) ⋊ C7 |
| Hall | C22×C127⋊C7 ⋊ C32 |
| Hall | C127⋊C7 ⋊ C3×A4 |
| Sylow | C127 ⋊ C3×C7×A4 |
| Sylow | C22 ⋊ C3×(C127⋊C3)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C22⋊((C127⋊C3)⋊C7) |
| commutator, cosocle, hypocenter | C22×C127 | C32×C7 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C127 | C3×C7 |
| LCS | G ▷ C22×C127 | |
| UCS | C1 ◁ C3 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1524 |
156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 172);