Finite Group DatabasebetaFinite Group Databasebeta| order | 5580 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C9 · Schur multiplier (H2) = C2 |
| direct product | C22⋊C9×ΓL(1,𝔽32) |
| unambiguous semidirect product | C22⋊(C9×ΓL(1,𝔽32)) |
| unambiguous semidirect product | (C31×C22⋊C9)⋊C5 |
| derived, Hall | C22×C31 ⋊ C5×C9 |
| Hall | C31×C22⋊C9 ⋊ C5 |
| Hall | C22×ΓL(1,𝔽32) ⋊ C9 |
| Sylow | C31 ⋊ C5×C22⋊C9 |
| Sylow | C22 ⋊ C9×ΓL(1,𝔽32) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A4×ΓL(1,𝔽32) |
| commutator, hypocenter | C22×C31 | C5×C9 |
| Fitting, socle | C22×C3×C31 | C3×C5 |
| LCS | G ▷ C22×C31 | |
| UCS | C1 ◁ C3 | |
32 conjugacy classes of subgroups.
132 conjugacy classes — view the full character table.
| maximal subgroup of | C5×C22⋊C9×ΓL(1,𝔽32) |
| maximal normal subgroup of | C5×C22⋊C9×ΓL(1,𝔽32) |
| maximal quotient of | C22⋊C9×C31⋊C25, C5×C22⋊C9×ΓL(1,𝔽32) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5580, 23);