Finite Group DatabasebetaFinite Group Databasebeta| order | 5580 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×A4×ΓL(1,𝔽32) |
| unambiguous semidirect product | C22⋊(C32×ΓL(1,𝔽32)) |
| unambiguous semidirect product | (C22×C3)⋊(C3×ΓL(1,𝔽32)) |
| unambiguous semidirect product | (C3×A4×C31)⋊C5 |
| Fitting, socle | C22×C3×C31 ⋊ C3×C5 |
| derived, Hall | C22×C31 ⋊ C32×C5 |
| Hall | C3×A4×C31 ⋊ C5 |
| Hall | C22×ΓL(1,𝔽32) ⋊ C32 |
| Sylow | C31 ⋊ C3×C5×A4 |
| Sylow | C22 ⋊ C32×ΓL(1,𝔽32) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4×ΓL(1,𝔽32) |
| commutator, hypocenter | C22×C31 | C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C31 | C3×C5 |
| LCS | G ▷ C22×C31 | |
| UCS | C1 ◁ C3 | |
56 conjugacy classes of subgroups.
132 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×A4×ΓL(1,𝔽32), C3×A5×ΓL(1,𝔽32) |
| maximal normal subgroup of | C3×C5×A4×ΓL(1,𝔽32) |
| maximal quotient of | C3×A4×C31⋊C25, C3×C5×A4×ΓL(1,𝔽32) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5580, 101);